Eigenspace decomposition of a linear involution ($P_{\pm }=(I\pm F)/2$)

Let $F$ be a linear map on a vector space over a field of characteristic different from two, with $F^2=I$. ThenThe projections satisfy $P_{+}{P}_{-}=0$ and $P_{+}+P_{-}=I$, giving existence and uniqueness of the decomposition. In characteristic two, the two signs coincide and the conclusion need not hold.