I have been working with something related to Goldman bracket for $G\_2$ gauge group. There I have something like "$\text{Tr}(M\_{\gamma}O\_i)$", where $M\_{\gamma}$ is a monodromy which takes value in the group $G\_2$ (in the fundamental representation) and $O\_i, i=1,2,\ldots,7$" are the 7x7 skew-symmetric matrices that can be obtained from the 8x8 matrix representation of left multiplication by octonionic imaginary units deleting the first row and the first column. 


My question is the following:
can $M\_{\gamma}O\_{i}M\_{\gamma}^{-1}$ be expressed as a suitable linear combination of the $O\_i$'s? The matrices $O\_i$'s are given explicitly in my paper:
<http://arxiv.org/pdf/1310.4519.pdf>
(eq. (4.3) p-19)



Action of G_2 on certain 7x7 skew-symmetric matrices
Yes. This follows from the fact that $G\_2$ acts on the octonions via automorphisms.


Let $e\_i$, $i=1,\dots,7$ be a choice of 7 imaginary octonion units and let $1$ denote the identity. Then by definition of the $O\_i$, we have that
$$
e\_i e\_j = \sum\_k e\_k (O\_i)\_{kj} - \delta\_{ij} 1
$$
where the left-hand side is octonion multiplication.


Let $g \in G\_2$. Since $G\_2$ acts on the octonions via automorphisms,
$$
g(e\_i) = \sum\_j e\_j M\_{ji}~.
$$
The $M\_{ji}$ are the entries of the matrix $M\_\gamma$ in the question.


Now apply $g \in G\_2$ on the first displayed equation. On the LHS we have
$$
g(e\_i e\_j) = g(e\_i) g(e\_j) = \sum\_{k,l} \left(\sum\_m e\_m (O\_k)\_{ml} - \delta\_{kl} 1\right) M\_{ki} M\_{lj} = \sum\_k (O\_k M)\_{mj} M\_{ki} - \delta\_{ij} 1~,
$$
where we have used that $M$ is orthogonal, so that $\sum\_{k,l} \delta\_{kl} M\_{ki} M\_{lj} = \delta\_{ij}$. Similarly, on the RHS we find
$$
\sum\_{m,k} e\_m M\_{mk} (O\_i)\_{kj} - \delta\_{ij} 1
$$
using that $g(1) = 1$.


Comparing the two, we find
$$
\sum\_{k,l,m} e\_m (O\_k)\_{ml} M\_{ki} M\_{lj} = \sum\_{m,k} e\_m M\_{mk} (O\_i)\_{kj}
$$
which is equivalent to
$$
\sum\_k (O\_k M)\_{mj} M\_{ki} = (M O\_i)\_{mj}
$$
or, finally,
$$
M O\_i M^{-1} = \sum\_k O\_k M\_{ki}~.
$$


**Edit** To answer the comment, there is a coordinate-free proof of this fact, and perhaps I should have written that first. What is $M O\_i M^{-1}$? It is the matrix of left multiplication by the imaginary octonions (projected to the imaginary octonions) under a $G\_2$ change of basis. But $G\_2$ is an automorphism of the octonions, hence it preserves the subspace of imaginary octonions. Hence $M O\_i M^{-1}$ is the matrix of left multiplication by *some* imaginary octonion unit, whence some linear combination of the $O\_i$, since these form a basis.


