Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian).
$L_X$ is a vector field on $M$, at a point $p$ it is so defined $$L_X(p)(f)=\left.\frac{d}{dt}\right\vert_{t=0}f(exp(-tX)\cdot p)$$
$L_X$ acts on $TM$ as the Lie derivative $L_X(Y)=[X,Y]$. Let $p\in M$ be such that $L_X(p)=0$, then the action of $L_X$ on $TM$ induces an action $L(X,p)$ on $T_pM$, it is linear.
Now comes the most technical part. Suppose $M$ has a $G$-invariant Riemannian metric. To my understanding, this means that the group multiplication is a local isometry.
Question (version 1)
The book I am reading says that from this follows that $L(X,p)$ is a skew symmetric operator. To be precise, it says $$L(X,p)\in\mathfrak{so}(T_pM)\;,$$Does that mean, or imply, what I wrote? In that case, or otherwise, how does one prove that membership?
If we define $$\alpha_g:M\rightarrow M\;;x\mapsto g\cdot x$$then $\alpha_g(p)=p$ so we have, differential geometry speaking, $d\alpha_g:T_pM\rightarrow T_pM$ is an isometry. The problem is I am not really sure how to relate this differential with the Lie derivative, are they precisely the same thing?. If they are, or if they are closely related, the only thing I can say about $d\alpha_g$ is that it is a member of $O(T_pM)$ which is quite different from $\mathfrak{so}(T_pM)$.
Question (version 2)
If you can answer version 1 you don't really need to read this. Maybe some of you already know where I am going with this. I need to show that I can put $L(X,p)$ in block diagonal form, with blocks of the form $$\begin{pmatrix}0&\lambda\\-\lambda&0\end{pmatrix}$$.
Based just on the background information, can you say there is a vector basis that would put $L(X,p)$ in that form?
Motivation
This is a step in proving Berline-Vergne localization theorem for isolated fixed points. See, for example, Berline-Getzler-Vergne, Heat Kernels and Dirac Operators or `Lecture notes on Equivariant cohomology' by Matvei Libine.