If $S$ and $T$ are projections with $S\circ T \neq T\circ S$, then can $S\circ T$ ever be a projection?

Question

I'm looking to find out if $S$ and $T$ are projections with $S\circ T \neq T\circ S$, then can $S\circ T$ ever be a projection. If it can what is an example of it? If it can't how would I go about proving it?

Answer 1

Let $W\leq V$ be any proper subspace and $W', W''\leq V$ two subspaces such that $V=W\oplus W'=W\oplus W''$. Let $T$ be the projection onto $W$ along $W'$ and $S$ the projection onto $W$ along $W''$. Then $TS=S$ and $ST=T$ are both projections, but $ST\neq TS$ since $\ker ST=W'$ and $\ker TS=W''$.

Can you construct an example $\mathbb{R}^2\to \mathbb{R}^2$?