Let $G$ be a finite abelian subgroup of $O_{2n+1}(\mathbb R)$. Suppose that $G\not\subset \{-I_n,I_n\}$.
There exists $\varepsilon \in \{-1,1\}$ and $M\in G$ such that $0<\dim \ker (M-\varepsilon I_n)<n$.
Since $G$ is abelian, $V:=\ker (M-\varepsilon I_n)$ is stable by $G$. Using a fairly elaborate result of simultaneous similarity I could state that there exists a subspace $W$ such that $V\oplus W =\mathbb R^n$ and $W$ is stable by $G$. However I'd like to find a direct (simple) proof of that last statement.
Now, if we replace $2n+1$ by $2n$, we can replace $V$ by a subspace of dimension 2, and I'd like a similar result/proof for $W$ (i.e. direct sum and stability) without using the elaborate result mentioned above.