= Solution
\b[The proposed equivalence is false.] On a nonzero <vector space> over any <field> of <characteristic> different from two, choose
$$
E=I,\qquad F=I,\qquad G=0.
$$
These <linear projections> commute and satisfy $EFG=0$. Nevertheless their sum is $2I$, and
$$
\boxed{(2I)^2-2I=2I\ne0.}
$$
Thus the proposed conditions are not sufficient. For commuting <linear projections> the actual obstruction is the sum of the pairwise products:
$$
(E+F+G)^2-(E+F+G)=2(EF+EG+FG).
$$
The single condition on the triple product does not remove this obstruction.
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