Is the sum of two projections a projection? Let $ S $ and $ T $ be two linear subspaces of $ \Bbb{R}^{2} $. Then is the sum of the projections $ P_{S} $ and $ P_{T} $ (i.e., $ P_{S} + P_{T} $) a projection?
I don’t think it is since the projection rule doesn’t hold ($ P_{M}^{2} $ does not equal $ P_{M} $), but I was hoping someone could solidify this. It would be very much appreciated!

The sum is a projection iff $$
p_1 \circ p_2 = - p_2 \circ p_1
$$
because
\begin{align}
(p_1 + p_2)\circ (p_1 + p_2) - (p_1 + p_2) &=
\color{red}{p_1 \circ p_1} + p_1 \circ p_2 + p_2 \circ p_1 
+  \color{blue}{p_2 \circ p_2}
- \color{red}{p_1} - \color{blue}{p_2} \\&=
p_1 \circ p_2 + p_2 \circ p_1
\end{align}
