  For a representation of a connected compact Lie group G in a finite
dimensional real vector space U and a subspace V of U, invariant under a
maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits
in U, which is also necessary in certain cases. The proof makes use of the
cohomology of flag manifolds and the invariant theory of Weyl groups. Then we
apply our condition to the conjugation representations of U(n), Sp(n), and
SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In
particular, we obtain an interesting generalization of Schur's
triangularization theorem.
