Suppose $F$ is a field where $1 \neq -1$ and $V$ is a $2n$ dimensional $F$-vector space. Also suppose that $M,N$ are involutions, i.e. $M^2 = I$ and $N^2 = I$, and that $M$ and $N$ anti-commute, i.e. $MN = -NM$.
I would like to show that $ M = \left[\begin{array}{cc}A & 0\\ 0 & -A \end{array} \right],~~ N =\left[\begin{array}{cc}0 & B\\ B & 0 \end{array} \right] $ (these are given in block matrix notation, so that $A,B$ are matrices, not scalars).
I found this website which makes the makes the same claim ($M,N$ involutions implies they are invertible, the website handles a slightly more general case) but I am not able to follow the argument.
First off, could anyone verify that this is true for an arbitrary $F$-vector space as described? Also some help with the proof would be much appreciated, thank you.