Existence and construction of two-dimensional invariant subspaces for pairs of rotations
By a rotation in a Euclidean space $V$ of even dimension we mean an orthogonal linear operator on $V$ which is an orthogonal direct sum of rotations in 2-dimensional linear subspaces of $V$ by a common angle $\alpha \in [0,\pi ]$. We present a criterion for the existence of a 2-dimensional subspace of $V$ which is invariant under a given pair of rotations, in terms of the vanishing of a determinant associated with that pair. This criterion is constructive, whenever it is satisfied. It is also used to prove that every pair of rotations in $V$ has a 2-dimensional invariant subspace if and only if the dimension of $V$ is congruent to 2 modulo 4.