 Solution

Let $P$ have columns $a,b,c$ and let $G$ have columns $A,B,C$. The cross-product formulas defining the reciprocal basis giveprovided the original scalar triple product $T=\det P$ is nonzero. Taking determinants yields the reciprocal basis involution identityWhen $T=0$, the displayed reciprocal vectors are undefined, so there is no additional zero-denominator case to evaluate.