#
Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Modeling with Lie Groups

^{23}, where SE(3) is the Lie group of rigid body transformations in 3D, i.e., the Lie group of rotations and translations in ${\mathbb{R}}^{3}$, also called the special Euclidean group.

#### 1.2. Statistics on Lie Groups

_{i}}

_{i}

_{=1}, …,

_{N}is a solution, if there are some, of the following group barycenter equation:

#### 1.3. Using Riemannian and Pseudo-Riemannian Structures for Statistics on Lie Groups

#### 1.4. Lie Groups and Lie Algebras with Bi-Invariant Pseudo-Metrics

^{*}-extension structure [23]. They have been completely described for certain dimensions in specific cases. The classification of the nilpotent quadratic Lie algebras of dimensions ≤ 7 is obtained in [24], of the real solvable quadratic Lie algebras of dimensions ≤ 6 in [25] and the irreducible non-solvable Lie algebras of dimensions ≤ 13 in [26]. The specific cases of indecomposable quadratic Lie algebras with pseudo-metrics of different indices have been studied: bi-invariant pseudo-metrics of index one are described in [21,27], of index two in [28] and finally of the general index in [29]. The dimension of the space of bi-invariant pseudo-metrics has been studied in [30] where bounds are provided.

#### 1.5. Contributions and Outline

## 2. Introduction to Lie Groups with Bi-Invariant Pseudo-Metrics

#### 2.1. Quadratic Lie Groups and Lie Algebras

#### 2.1.1. Lie Groups

_{h})(X(g)) = X(L

_{h}(g)) = X(h * g) for each g, $h\in \mathcal{G}$. Similarly, one could define right invariant vector fields. The left invariant vector fields form a vector space that we denote $\mathrm{\Gamma}{(T\mathcal{G})}^{L}$ and that is isomorphic to ${T}_{e}\mathcal{G}$. The Lie bracket of two left invariant vector fields is a left-invariant vector field [39].

#### 2.1.2. Lie Algebras

_{g}on a given basis ${\mathcal{B}}_{\mathfrak{g}}={\left\{{e}_{i}\right\}}_{i=1}^{n}$ of $\mathfrak{g}$, we define the structure constants f

_{ijk}as:

_{ijk}depend on the basis ${\mathcal{B}}_{\mathfrak{g}}$ chosen. They are always skew-symmetric in the first two indices, but they may have additional symmetry properties if we write them in a well-chosen basis (see below). The structure constants f

_{ijk}completely determine the algebraic structure of the Lie algebra. Therefore, the structure constants are often the starting point, or the input, of algorithms on Lie algebras [34–36,38]. It will also be the case for the algorithm we present in this paper.

#### 2.1.3. Pseudo-Metrics

_{g}on each tangent space ${T}_{g}\mathcal{G}$. Then, $\mathcal{G}$ becomes a pseudo-Riemannian manifold. A metric is defined as a pseudo-metric whose inner products are all positive definite. In this case, $\mathcal{G}$ is called a Riemannian manifold.

_{g}at a point g and with respect to a basis of ${T}_{g}\mathcal{G}$. The signature is independent of the choice of the point g and on the basis at ${T}_{g}\mathcal{G}$. By definition, a pseudo-metric is definite; thus, there are no null eigenvalues, and we have p + q = n, where n is the dimension of $\mathcal{G}$. By definition, a metric is positive definite, and thus, its signature is (n, 0). Again, further details about such differential geometry can be found in [39].

#### 2.1.4. Quadratic Lie Groups and Algebras

_{h}is the left translation by h. In other words, the left translations are isometries for this pseudo-metric. Similarly, we can define right-invariant and bi-invariant pseudo-metrics <, >. Note that any Lie group admits a left (or right) invariant pseudo-metric: we can define an inner product on the Lie algebra $\mathfrak{g}={T}_{e}\mathcal{G}$ and propagate it on each tangent space ${T}_{g}\mathcal{G}$ through DL

_{g}(e) (or DR

_{g}(e)). However, no Lie group admits a bi-invariant pseudo-metric.

#### 2.1.5. Characterization of Quadratic Lie Algebras

_{i}, y = e

_{j}and z = e

_{k}. Thus, we can express the Lie bracket in terms of the structure constants, and we get:

_{i}) = A

_{i}, we can again reformulate Equation (4), and we get:

#### 2.1.6. How to Compute Bi-Invariant Pseudo-Metrics?

#### 2.2. Lie Algebra Representations

#### 2.2.1. Lie Algebras Representations

_{1}and θ

_{2}are said to be isomorphic if there is an isomorphism of representations between them, i.e., an isomorphism of vector spaces $l:{V}_{1}\mapsto {V}_{2}$ that verifies: θ

_{2}(x) ◦ l = l ◦ θ

_{1}(x). We denote ${\mathrm{Hom}}_{\mathfrak{g}}\left({V}_{1},{V}_{2}\right)$ the vector space of isomorphisms of representations between V

_{1}and V

_{2}.

#### 2.2.2. Adjoint and Co-adjoint Representation

_{i}defining the adjoint representation is equivalent to the set of structure constants of $\mathfrak{g}$.

#### 2.2.3. Some Vocabulary of Algebra

#### 2.2.4. Some Vocabulary of Geometry

^{⊥}≠ {0}. The ideal I is said to be totally isotropic if I ⊂ I

^{⊥}. The intersection between I and I

^{⊥}represents the vectors that are orthogonal to themselves and, thus, that have zero norm, even if they are themselves non-zero.

^{⊥}as photons: they have zero mass even if they have non-zero velocity.

#### 2.3. Constructions with Lie Algebra Representations

_{B}) to denote the Lie algebra, because this is the notation that we will use in the core of our algorithm (see Section 4).

#### 2.3.1. Definition of Direct Sum

_{1}⊕

_{B}B

_{2}is the direct sum of B

_{1}, B

_{2}if:

- B = B
_{1}⊕ B_{2}in terms of vector spaces, - [B, B
_{1}]_{B}⊂ B_{1}and [B, B_{2}]_{B}⊂ B_{2}, making B_{1}and B_{2}subrepresentations of the adjoint representation of B, in other words: ideals of B.

_{1}and B

_{2}i.e.,:

_{1}⊕

_{B}B

_{2}. Note that we write ⊕

_{B}to emphasize the fact that this direct sum decomposition is more than the direct sum decomposition into vector spaces.

#### 2.3.2. Direct Sum Decomposition and Bi-Invariant Pseudo-Metrics

_{1}and B

_{2}being quadratic. Indeed, if <, >

_{B}

_{1}, <, >

_{B}

_{2}are bi-invariant pseudo-metrics on B

_{1}, B

_{2}and represented by the matrices ${Z}_{{B}_{1}}$, ${Z}_{{B}_{2}}$, then:

_{B}is bi-invariant on B, its restrictions <, >

_{B}|

_{B}

_{1}and <, >

_{B}|

_{B}

_{2}are bi-invariant on B

_{1}, B

_{2}[20,21].

#### 2.3.3. Computing the Direct Sum

_{k}= A(e

_{k}), computing the direct sum decomposition of B into indecomposable B

_{i}’s amounts to the simultaneous bloc diagonalization of the matrices A

_{k}.

#### 2.3.4. Definition of Double Extension

- B = W ⊕ S ⊕ S
^{*}in terms of vector spaces, - (W, [, ]
_{W}) is a Lie algebra and [S, W ]_{B}⊂ W makes W a S-representation, - (S, [, ]
_{S}) is a simple Lie subalgebra of B: [s, s′]_{B}= [s, s′]_{S}, - S* is the dual space of S and [S, S*]
_{B}⊂ S* makes S* the co-adjoint representation, - ∀w, w′ ∈ W : [w, w′]
_{B}= [w, w′]_{W}+ β(w, w′) where β : Λ^{2}W ↦S* is a (skew-symmetric)

_{B}of B, i.e., the matrices denoted: b ↦A(b). The double extension decomposition is equivalent to the following decomposition of the adjoint representation of B:

_{S}and ad(w) = [w, •]

_{W}to be respectively the adjoint representation of S (on S) and the adjoint representation of W (on W ). However, [s, •]

_{B}is a S-representation on W that has nothing to do with the adjoint (the adjoint is a representation of a Lie algebra on itself).

#### 2.3.5. Double Extension Decomposition and Bi-Invariant Pseudo-Metrics

_{W}is bi-invariant on W, represented by Z

_{W}, then:

_{W}=<, >

_{B}|

_{W}is bi-invariant [20,21]. Note here that we can write the II-blocks, because the basis of S and S

^{*}are chosen to be duals of each other. If two different basis were chosen, the corresponding bi-invariant pseudo-metric on B = W ⊕ S ⊕ S

^{*}would have the form:

_{W}

_{⊕}

_{S}

_{⊕}

_{S}* while choosing s ∈ S, we show that L is necessarily an isomorphism of S-representations on S and I, i.e., L ∈ Hom

_{S}(S, S*). This remark will be used in practice in the algorithm (see Section 4).

#### 2.3.6. Computing Double Extensions

^{⊥}its orthogonal with respect to a bi-invariant pseudo-metric <, >

_{B}. The decomposition:

^{⊥}verify the following properties:

- I is abelian,
- I
^{⊥}is a maximal ideal, - I ⊂ I
^{⊥}(total isotropy), - [I, I
^{⊥}] = 0 (commutativity), - codim(I
^{⊥}) = dim(I).

^{⊥}needed for the construction shown above: we do not know any bi-invariant pseudo-metric, as we want to build one!Thus, given an abelian minimal ideal I, we shall test all ideals J that could be an I

^{⊥}for a bi-invariant pseudo-metric, i.e., all ideals J that verify the necessary conditions listed above.

^{⊥}are either J = C

_{B}(I) the centralizer of I in B in the case C

_{B}(I) ≠ B or the maximal ideals of codimension one containing I in the case C

_{B}(I) = B.

^{⊥}is its commutativity with I: [I, J] = 0. We recall that the centralizer C

_{B}(I) of I in B is defined as the set of elements that commute with I. Thus: J ⊂ C

_{B}(I).

_{B}(I) is also an ideal. Thus, J is a maximal ideal included in the ideal C

_{B}(I): we have necessarily J = C

_{B}(I) in the case C

_{B}(I) ≠ B. In this case, the condition I ⊂ J is fulfilled as I is abelian. The last necessary condition to check is codim(C

_{B}(I)) = dim(I).

_{B}(I) = B, then we shall look for maximal ideals of B. However, in this case, I commutes with all elements of B, and therefore, I is necessarily of dimension one as a minimal ideal. Therefore, we shall look for maximal ideals J of codimension one. Adding the last necessary condition, we conclude that in the case C

_{B}(I) = B, we shall consider only maximal ideals of codimension one containing I.

## 3. Structure of Quadratic Lie Groups

#### 3.1. A Classification Theorem

**Theorem 1**(Classification of quadratic Lie algebras). The Lie algebra$\mathfrak{g}$ is quadratic if and only if its adjoint representation decomposes into indecomposable subrepresentations B that are of the following types:

- Type (1): B is simple (or one-dimensional),
- Type (2): B = W ⊕ S ⊕ S* is a double extension of a quadratic W by S simple (or one-dimensional).

#### 3.1.1. Elementary Bi-Invariant Pseudo-Metrics

_{B}of a one-dimensional Lie algebra B is defined to be the multiplication. The elementary bi-invariant pseudo-metric <, >

_{B}of a simple Lie algebra B is defined to be the Killing form. Now, let us define recursively the elementary bi-invariant pseudo-metrics of a general quadratic $\mathfrak{g}$.

_{B}as above: the multiplication if B is one-dimensional or the Killing form if B is simple.

^{⊥}. We get the double extension B = W ⊕ S ⊕ S

^{*}with W = I

^{⊥}/I, S = B/I

^{⊥}and S* = I. We construct an elementary bi-invariant pseudo-metric <, >

_{W}on W recursively. We then define an elementary bi-invariant pseudo-metric <, >

_{B}on the double extension B = W ⊕ S ⊕ S* to be of the form of Equation (14).

#### 3.2. Riemannian and Pseudo-Riemannian Quadratic Lie Groups

#### 3.2.1. Studying the Signature

_{i}are either simple (or one-dimensional) or double extensions. The signature on the direct sum is the sum of the signatures on the B

_{i}[39]:

#### 3.2.2. Comparison

#### 3.3. From a Bi-Invariant Pseudo-Metric to a Bi-Invariant Dual Metric?

#### 3.3.1. Dual Numbers and Vectors

^{2}= 0 and ϵ ≠ 0 defines the multiplication [44]. We can define an m-dimensional dual vector space ${\mathbb{D}}^{m}={\mathbb{F}}^{m}+\u03f5{\mathbb{F}}^{m}$, whose elements are dual vectors. Note here that the term “vector” is abusive in the sense that a vector space is usually defined on a field, not on an algebra. In the following, in order to study the properties of the dual vector space, we will use the dual map:

#### 3.3.2. From the Double Extension $\mathfrak{g}=K\oplus K*$ to Its Dual $\overline{\mathfrak{g}}=K+\u03f5K*$

_{0}∈ K and x

_{ϵ}∈ K*.

**Proposition 1**. The dual map:

_{K}

_{⊕}

_{K}* above.

_{K}

_{⊕}

_{K}* on the Lie $\mathbb{F}$-algebra $\mathfrak{g}$ maps to the canonical metric $Z=\mathbb{I}$ on the Lie $\mathbb{D}$-algebra $\overline{\mathfrak{g}}$:

#### 3.3.3. Towards Statistics on Dual Riemannian Manifolds

#### 3.3.4. Generalization?

^{2}= 0, enables one to represent the commutativity of K* (Lie bracket is null) and the self-orthogonality of K* (the inner product is null) at the same time. A general Lie algebra with the bi-invariant pseudo-metric is not necessarily decomposable into two subspaces of same dimension, such that one of them is abelian and isotropic. For example, take a Lie algebra of an odd dimension.

## 4. An Algorithm to Compute Bi-Invariant Pseudo-Metrics on a Given Lie Group

#### 4.1. The Algorithm: Computation of One Bi-Invariant Pseudo-Metric

_{ijk}on this basis. The output is a symmetric invertible matrix ${Z}_{\mathfrak{g}}$ on the basis ${\mathfrak{B}}_{\mathfrak{g}}$, representing an elementary bi-invariant pseudo-metric, or a message of error: “the Lie algebra $\mathfrak{g}$ is not quadratic”.

#### 4.1.1. Core of the Algorithm

- B is of Type (2),
- B is quadratic,
- ∀I minimal, I abelian, there is a double extension decomposition of B,
- ∃I minimal, abelian, such that there is a double extension decomposition of B.

_{B}(I), the maximal ideals J’s and the corresponding candidates for the double extension structure of B. The computation of C

_{B}(I) is implemented in [47].

_{B}(I) ≠ B, we take J = C

_{B}(I) and verify the condition codim(J) = dim(I). If the condition is not fulfilled, there is no double extension structure possible for B. Therefore, we conclude that B is not of Type (2).

_{B}(I) = B, we compute the maximal ideals J of B of codimension one containing I (see Section 2). If no such ideals are found, there is no double extension structure possible for B. Again, in this case, we conclude that B is not of Type (2).

_{W}).

_{W}, S = B/J and S* = I. Given a double extension candidate, we know from Section 2 that an elementary pseudo-metric on B has the form:

_{S}(S, I).

_{S}(S, I). We recall that S is simple; thus, its adjoint representation is irreducible. As we are in the case of a finite dimensional irreducible representation, we can apply Schur’s lemma. Its general form states that Hom

_{S}(S, S) is an associative division algebra over $\mathbb{F}\left(=\mathbb{R}\phantom{\rule{0.2em}{0ex}}or\phantom{\rule{0.2em}{0ex}}\mathbb{C}\right)$, which is of finite degree, because S is finite dimensional [48]. When the base field is $\mathbb{F}=\mathbb{C}$, we use the fact that a finite-dimensional division algebra over an algebraically closed field is necessarily itself. Thus, ${\mathrm{Hom}}_{S}\left(S,S\right)=\mathbb{C}$ and ${\mathrm{dim}}_{\mathbb{C}}\left({\mathrm{Hom}}_{S}\left(S,S\right)\right)=1$. When the base field is $\mathbb{F}=\mathbb{R}$, we use the Frobenius theorem, which asserts that the only real associative division algebras are $\mathbb{R}$, $\mathbb{C}$ or $\mathrm{H}$, the field of quaternionnumbers [49]. Thus, Hom

_{S}(S, S) is $\mathbb{R}$, $\mathbb{C}$ or $\mathrm{H}$, and dim

_{R}(Hom

_{S}(S, S)) is 1, 2 or 4. Now, if I and S are isomorphic, Hom

_{S}(S, I) is isomorphic to Hom

_{S}(S, S) and, thus, of maximal dimension four over $\mathbb{F}$. Otherwise, if I and S are not isomorphic, we have Hom

_{S}(S, I) = {0}.

_{S}(S, I) is implemented in [50], more generally for any finite-dimensional modules of a finitely generated algebra.

_{B=W}

_{⊕}

_{S}

_{⊕}

_{I}into Equations (7) and solve it for L. Thus, the initial system of Equations (7) has been reduced to an equation in maximum one (complex case) or in four (real case) parameters.

_{B}is found on one of the candidates, we return Z

_{B}. Otherwise, we conclude that B is not of Type (2).

_{B}’s that have been returned on the B’s.

#### 4.1.2. Tree Structure of the Algorithm

_{g}(e) (or DR

_{g}(e)) on all tangent spaces ${T}_{\mathfrak{g}}\mathcal{G}$ (see Section 2).

#### 4.2. Generalization of the Algorithm: Computation of All Bi-Invariant Pseudo-Metrics

#### 4.2.1. Computing the Quadratic Space of Indecomposable Lie Algebras

#### 4.2.2. Computing the Quadratic Space of a Direct Sum

_{i}. This gives:

_{ij}is a matrix that solves the following equation, derived from Equations (7):

#### 4.3. Results of the Algorithm on Selected Lie Groups

#### 4.3.1. Scalings and Translations ST (n)

_{1}, t

_{1}) * (λ

_{2}, t

_{2}) = (λ

_{1}.λ

_{2}, λ

_{1}* t

_{2}+ t

_{1}) and (λ, t)

^{(−1)}= (1/λ, −t/λ).

_{a}= (0, e

_{a}) with ${({e}_{a})}_{a=1}^{n}$ the canonical basis of ${\mathbb{R}}^{n}$. In this basis, the structure constants can be read in the following Lie brackets:

_{1}), …, Span(P

_{n}) and their linear combinations. We remark that there is no ideal containing D. Thus, $\mathfrak{st}(n)$ cannot be written as the direct sum of ideals, i.e., $\mathfrak{st}(n)$ is indecomposable.

_{1}), for example, is an ideal, $\mathfrak{st}(n)$ is not simple. We conclude that $\mathfrak{st}(n)$ is not of Type (1).

_{1}), which is obviously a minimal abelian ideal. From the commutation relations given by the Lie brackets, we see that ${C}_{\mathfrak{st}(n)}(I)=Span({\left\{{P}_{a}\right\}}_{a=1}^{n})$, and we are in the case ${C}_{\mathfrak{st}(n)}(I)\ne \mathfrak{st}(n)$. Thus, there is only one double extension candidate, with $J={C}_{\mathfrak{st}(n)}(I)$. We define $S=\mathfrak{st}(n)/J=\text{Span}(D)$ and W = J/I = Span(P

_{2},..P

_{n}). We call the algorithm recursively on W, which decomposes into one-dimensional ideals on which we return the multiplication.

_{1}] = P

_{1}. Hence, I and S are not isomorphic S-representations, and Hom

_{S}(S, I) is zero. We conclude that $\mathfrak{st}(n)$ is not of Type (2).

#### 4.3.2. Heisenberg Group H

_{1}, y

_{1}, z

_{1}) * (x

_{2}, y

_{2}, z

_{2}) = (x

_{1}+ x

_{2}, y

_{1}+ y

_{2}, z

_{1}+ z

_{2}+ x

_{1}* y

_{2}) and group inversion (x, y, z)

^{(−1)}= (−x, −y, −z + xy).

_{W}= 1.

_{S}(S, I) is obviously one.

#### 4.3.3. The Group of Scaled Upper Unitriangular Matrices UT (n)

#### 4.3.4. Rigid Body Transformations SE(n)

_{1}, t

_{1}) * (R

_{2}, t

_{2}) = (R

_{1}.R

_{2}, R

_{1}* t

_{2}+ t

_{1}) and (R, t)

^{(−1)}= (R

^{(−1)}, R

^{(−1)}.(−t)).

_{se(}

_{n}

_{)}(I) = P = I. The necessary condition codim(J) = dim(I) is verified only for n = 3. We conclude that $\mathfrak{se}$(n) is not of Type (2) if n ≠ 3. We go on with n = 3. We compute S = $\mathfrak{se}$(3)/P ∼ $\mathfrak{so}$(3) and W = P/P = {0}.

_{1}= J

_{23}, J

_{2}= J

_{31}and J

_{3}= J

_{12}. The S-representation on S is the adjoint representation: [J

_{m}, J

_{n}] =

_{mnp}.J

_{p}. The S-representation on I = P is given by: [J

_{m}, P

_{a}] =

_{map}.P

_{p}. It is also the adjoint representation. The isomorphism of vector spaces L that maps each P

_{a}on J

_{a}is an isomorphism of representations whose matricial form is the identity in our basis.

_{se(3)}on the decomposition S ⊕ I = $\mathfrak{so}$(3) ⊕ P with basis $({\left\{{J}_{a}\right\}}_{a=1}^{3},{\left\{{P}_{a}\right\}}_{a=1}^{3})$ and get:

_{se(3)}is bi-invariant on $\mathfrak{se}$(3). Z

_{se(3)}is actually known as the Klein form [51].

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Left and right translation of a dataset ${\left\{{g}_{i}\right\}}_{i=1}^{N}$ on the Lie group G. The initial dataset ${\left\{{g}_{i}\right\}}_{i=1}^{N}$ has a mean represented in red. The left translated dataset ${\{h*{g}_{i}\}}_{i=1}^{N}$ has a mean represented in blue. The right translated dataset ${\{{g}_{i}*h\}}_{i=1}^{N}$ has a mean represented in green. We require that the mean of the (right or left) translated dataset is the translation of the red mean, which is the case in this illustration: the blue mean is the left translation of the red mean, and the green mean is the right translation of the red mean.

**Figure 2.**Algebraic and geometric structures. If we require compatible algebraic and geometric structures on the manifold, we get a quadratic Lie group: a Lie group with a bi-invariant pseudo-metric.

**Figure 6.**Schematical result for ST (n). We see on the top level that $\mathfrak{st}(n)$ is indecomposable (it decomposes into itself). We see on the bottom level that $\mathfrak{st}(n)$ is neither one-dimensional, nor simple, nor a double extension, and therefore, we exit the algorithm: $\mathfrak{st}(n)$ is not quadratic.

**Figure 7.**Schematical result for H and UT (n). The top level indicates the direct sum decomposition step. Thus, $\mathfrak{h}$ is indecomposable, and $\mathfrak{ut}\left(n\right)$ decomposes into $\mathfrak{d}$ and $\mathfrak{h}$. The bottom level for $\mathfrak{h}$ indicates that $\mathfrak{h}$ is neither one-dimensional, nor simple, nor a double extension, and therefore, we exit the algorithm: $\mathfrak{h}$ is not quadratic. The bottom level for $\mathfrak{ut}\left(n\right)$ indicates that $\mathfrak{d}$ is one-dimensional and therefore quadratic, but that $\mathfrak{h}$ is not quadratic: $\mathfrak{ut}\left(n\right)$ is not quadratic.

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Miolane, N.; Pennec, X.
Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics. *Entropy* **2015**, *17*, 1850-1881.
https://doi.org/10.3390/e17041850

**AMA Style**

Miolane N, Pennec X.
Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics. *Entropy*. 2015; 17(4):1850-1881.
https://doi.org/10.3390/e17041850

**Chicago/Turabian Style**

Miolane, Nina, and Xavier Pennec.
2015. "Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics" *Entropy* 17, no. 4: 1850-1881.
https://doi.org/10.3390/e17041850