= Eigenspaces of a symplectic involution
{title2=$V=V_+\oplus V_-$}
If $T$ preserves a <nondegenerate bilinear form> $\omega$ that is alternating and satisfies $T^2=I$ on a finite-dimensional <vector space> over a <field> of characteristic different from two, then
$$
V=\ker(T-I)\oplus\ker(T+I).
$$
The two <eigenspaces> are orthogonal for $\omega$, since $\omega(v_+,v_-)=\omega(Tv_+,Tv_-)=-\omega(v_+,v_-)$. Each restricted form is nondegenerate: a vector annihilating its own eigenspace also annihilates the other and hence all of $V$. Both dimensions are even, say $2a$ and $2b$. Consequently $\operatorname{tr}T=2a-2b=\dim V-4b$ as an integer obtained from the eigenspace dimensions. Over the real numbers this gives the congruence $\operatorname{tr}T\equiv\dim V\pmod4$.
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