 Invariant subspaces of a quarter-turn in three dimensions

For a quarter-turn rotation $T$ of ${\mathbb{R}}^3$ about a unit axis vector $n$, the only real invariant subspaces are $\{0\}$, the axis $\mathbb{R}n$, its perpendicular plane $n^{\perp }$, and all of ${\mathbb{R}}^3$.

Indeed $T^2$ fixes the axis and negates the perpendicular plane. Hence $P=(I+T^2)/2$ is the linear projection onto the axis, while $I-P$ is the linear projection onto the perpendicular plane. An invariant subspace $L$ is preserved by these polynomials in $T$, soThe first intersection is either zero or the axis. If the second contains a nonzero $v$, it also contains $Tv$. These are perpendicular nonzero vectors, so they span the perpendicular plane. This proves the classification.

Over the complex numbers the situation changes: the rotation matrix has eigenvalues $1,i,-i$, giving three one-dimensional eigenspaces. The absence of real eigenvalues on the perpendicular plane explains why it has no real one-dimensional invariant subspace.