Action of G_2 on certain 7x7 skew-symmetric matrices 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)
 A: 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.
