Polynomial reduction of a slashed vector ($F(/p)=A(p^2)/p+B(p^2)I$)

The Clifford algebra gives $(/p{)}^2=p^{2}I$. Splitting a scalar-coefficient polynomial into even and odd powers proves the displayed reduction. For an on-shell fermion spin sum assumed to be such a polynomial, $(/p-m)P=0$ implies $P=A(/p+m)$; the normalization $\mathrm{tr}(P{\gamma }^0)=4p^0$ fixes $A=1$ in four spacetime dimensions.