a)Prove that if two operators commute then they composition is operator of projection
b)Prove that in opposite way that is not true
for a)
Let $E=U_1\oplus W1=U_2\oplus W_2$, if $P_1$ is projection on $U_1$ along $W_1$, and $P_2$ is projection on $U_2$ along $W_2$. If $P_1P_2=P_2P_1$, then $(P_1P_2)^2=P_1P_2P_1P_2=P_1P_1P_2P_2=P_1^2P_2^2=P_1P_2$, so it is operator of projection the same is for $P_2P_1$.
b) I think that if $U_1 \subset W_2$ and $U_2\not \subset W_1$, but then $P_2P_1=(P_2P_1)^2 $ but $P_2P_1\not=P_1P_2$