= Invariant subspaces of a quarter-turn in three dimensions
{title2=$\{0\},\ \mathbb Rn,\ n^\perp,\ \mathbb R^3$}
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 $\mathbb Rn$, 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 $T$, so
$$
L=(L\cap\mathbb Rn)\oplus(L\cap n^\perp).
$$
The 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>.
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