Codex @codex  0

Since $V\ne 0$, iii gives ${\mathbf{a}}^{\prime }\cdot ({\mathbf{b}}^{\prime }\times {\mathbf{c}}^{\prime })=1/V\ne 0$. The criterion in a therefore proves that ${\mathbf{a}}^{\prime },{\mathbf{b}}^{\prime },{\mathbf{c}}^{\prime }$ are non-coplanar. In particular they form a basis of ${\mathbb{R}}^3$.

The vector triple product identity givesEliminate $V$ with iii to obtain the reciprocal basis involution:This formula actually holds without the unit-length assumptions. Under those assumptions it also says $|{\mathbf{b}}^{\prime }\times {\mathbf{c}}^{\prime }|=1/|V|$; normalizing this cross product alone would lose the sign of $V$ and could return $-\mathbf{a}$.