Consider the real Lie algebra $\mathfrak{so}_{p,q}(\mathbb{R})$ with $p+q=m$, we label the elements of the algebra $X_{i,j}$ for $1\leq i<j\leq m$. Let $Y$ be a set of $m$ vectors.
Can I always find a basis $(Y_1,\cdots,Y_m)$ of $Y$ such that
$$ [X_{i,j},Y_i]= -Y_j\quad \text{and}\quad [X_{i,j},Y_j] =\pm Y_i $$
and of course $[X_{i,j},Y_k]=0$ when $k\neq i,j$ ? The $\pm$ sign depend on the ratio of the eigenvalues of the underlaying quadratic form.
Observe that the above relations mean that $Y_i$ and $Y_j$ are eigenvectors of $[X_{i,j},[X_{i,j},\cdot]]$.
Thank you!
Btw do you have some references about real Lie algebras without going throught the complex field ?