Kernel of some expressions in real Lie algebras

Let $\mathfrak{g}$ be a real Lie algebra and let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition of $\mathfrak{g}$. Let $\mathfrak{a}$ be maximal abelian subalgebra of $\mathfrak{p}$. Then $\mathfrak{g}$ have a restricted root space decomposition \begin{align}\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\lambda\in\Phi}\mathfrak{g}_{\lambda},\end{align} where \begin{align}\mathfrak{g}_{\lambda}=\left\{ X\in\mathfrak{g}\::\:\forall H\in\mathfrak{a}\;\:\mbox{ad}\left(H\right)X=\lambda\left(H\right)X\right\}.\end{align}

The following are two statements which I have managed to prove but have not been able to find in the literature. A reference or citation will be much appreciated. Any literature in a broader context will also be very helpful.

Proposition 1. Let $\lambda,\mu\in\Phi, \lambda\neq\mu$, and $X_{\lambda}\in\mathfrak{g}_{\lambda},\: X_{\mu}\in\mathfrak{g}_{\mu}$. Assume $\mu-\lambda\notin\Phi$ and $\mu+n\lambda\in\Phi$ for $n\in\mathbb{N}$. If \begin{align}\mbox{ad}_{X_{\lambda}}^{n}X_{\mu}=0\end{align} then $X_{\lambda}=0$ or $X_{\mu}=0$.

Proposition 2. Assume $\mathfrak{g}$ is of type $C_{2}$. I.e. the positive roots are $\alpha,\beta,\alpha+\beta,\alpha+2\beta$. Let $X_{1}\in\mathfrak{g}_{\beta},\: X_{2}\in\mathfrak{g}_{\alpha+\beta},\: X_{3}\in\mathfrak{g}_{\alpha+2\beta}$ and $Y\in\mathfrak{g}_{-\alpha-2\beta}$. If \begin{align}\left[X_{3},Y\right]+\frac{1}{2}\left[X_{2},\left[X_{1},Y\right]\right]=0\end{align} then $X_{3}+\frac{1}{2}\left[X_{2},X_{1}\right]=0$ or $Y=0$.

• Since this is a reference request, this seems a reasonable question. Why the vote to close? – Yemon Choi May 8 '17 at 11:14