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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.

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  • $\begingroup$ I don't get the bit about how it follows from Schur's orthogonality relations. Could someone please explain? $\endgroup$ – Amritanshu Prasad Feb 10 '14 at 3:35
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    $\begingroup$ @AmritanshuPrasad: Write $w=e_0$ and extend to an orthonormal basis $(e_i)_{i<n}$. Let $u=\sum u_i e_i$ and same for $v$. Consider $\int (Ad_g u,e_0)(Ad_g v,e_0)$. Expanding the expression and using Schur leaves us with $n\sum u_i v_i=n(u,v)$. If this is negative, then for some $g$ the expression we are integrating is negative which gives what we want. $\endgroup$ – Pierre Simon Feb 11 '14 at 5:05

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