= Solution
For non-null <four-momentum>, the <transverse projector of a vector field> and the complementary <longitudinal projector of a vector field> are
$$
\boxed{P^a{}_b=\delta^a_b-\frac{q^aq_b}{q^2}},\qquad
L^a{}_b=\frac{q^aq_b}{q^2},\qquad q^2=\eta_{ab}q^aq^b\ne0.
$$
Indeed, $q_aP^a{}_b=0$, $P^2=P$, $L^2=L$, and $P+L=I$. Acting on an arbitrary <four-vector>,
$$
(P\widehat A)^a=\widehat A^a-q^a\frac{q_b\widehat A^b}{q^2},\qquad
(L\widehat A)^a=q^a\frac{q_b\widehat A^b}{q^2}.
$$
So $P$ removes the unwanted component, whereas $L$ extracts it. On the massive <mass shell>, $q^2=-m^2$ and $P^a{}_b=\delta^a_b+q^aq_b/m^2$. This on-shell form should not be used as an off-shell <linear projection>: away from $q^2=-m^2$ it is not idempotent. The non-null hypothesis matters; this decomposition is undefined at $q^2=0$.
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