= Sum of two linear projections
{title2=$(E+F)^2=E+F\Longleftrightarrow EF+FE=0$}
For <linear projections> $E,F$, expanding the square proves that their sum is a <linear projection> exactly when $EF=-FE$. Multiplying this equality on the left and right by $E$ gives $EF=-EFE$ and $EFE=-FE=EF$, hence $2EF=0$. If the <field> has <characteristic> different from two, this forces $EF=FE=0$, and those conditions also suffice. In <characteristic> two, the condition instead reduces to commutation; $E=F=I$ has sum zero but nonzero product. For three commuting <linear projections>, the square-minus-sum is $2(EF+EG+FG)$, so vanishing of the triple product alone is not sufficient.
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