Suppose we have a lie algebra with a $\mathbb{Z}_3$-decomposition: $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2$. Remember the relation $[\mathfrak{g}_i,\mathfrak{g}_j]\subset \mathfrak{g}_{i+j}$. As an imput, we have that $\mathfrak{g}_0\oplus\mathfrak{g}_2 = \mathfrak{su}(6)$ and $\mathfrak{g}=\mathfrak{so}(5)$. That is, the isometry group of the sphere and the stabilizer respectively. Also, $\mathfrak{g}_1$ has dimension $16$. Let $g$ be elements of the corresponding lie group. Now, consider the following operator:
$$ u \longmapsto (g^{-1}ug)_{even} $$
where $u$ is an element of $\mathfrak{g}_1$ and $(\cdots)_{even}$ meand the projection of $(\cdots)$ into the even part of the algebra $\mathfrak{g}_0\oplus\mathfrak{g}_2$.
My question is if I can prove the existence of an element $u\in \mathfrak{g}_1$ such that: $$(g^{-1}ug)_{even}\in\mathfrak{g}_0 $$ or $$ (g^{-1}ug)_{even}\in\mathfrak{g}_2 $$