Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector $w$. Given this data, can one always find an element $g\in G$ such that $Ad_g u$ and $Ad_g v$ fall on different sides of $H$? (In other words, $(Ad_g u,w)(Ad_g v,w)<0$.)

I know very little about Lie groups (this question came up in my research in model theory) and I am hoping that it might be easy for experts to answer.

Note that if the scalar product of $u$ and $v$ is negative, then the answer is yes and follows from Schur’s orthogonality relations.