Prove that a sum of projections is a projection iff they are orthogonal, if the characteristic of the space is not $2$ Let $E_1$ and $E_2$ be projections on $V$, a vector space over $F$. Why is if $\operatorname{char}F\neq2$ then $E_1+E_2$ is a projection iff $E_1E_2=E_2E_1=0$ ?

Easy direction: If $E_1E_2=E_2E_1=0$, then $(E_1+E_2)^2=E_1^2+E_2^2 = E_1+E_2$.  
Conversely, suppose $(E_1+E_2)^2= E_1+E_2$. Then 
$$
E_1E_2+E_2E_1 = 0 \tag{1}
$$
By multiplying both sides of (1) by $E_1$ on the left, or by $E_1$ on the right, obtain two equalities:
$$E_1 E_2+E_1E_2E_1=0,\qquad  E_1E_2E_1+E_2E_1=0$$
By subtracting these, 
$$E_1E_2-E_2E_1=0. \tag{2}$$
Adding or subtracting (1) and (2),  we get 
$$2E_1E_2=0,\qquad  2E_2E_1=0$$
Since the characteristic is not $2$, the factor $2$ can be cancelled.
(This answer is based on this post by Alex).
