Reciprocal basis involution ($[{\hat{e}}_1,{\hat{e}}_2,{\hat{e}}_3]=1/[e_1,e_2,e_3]$)

For a three-dimensional basis with nonzero scalar triple product $\Delta$, its reciprocal basis satisfies ${\hat{e}}_2\times {\hat{e}}_3=e_1/\Delta$ and the cyclic analogues. Consequently its scalar triple product is $1/\Delta$, and taking the reciprocal basis again recovers the original basis. The coordinates of $V$ in the original basis are $V\cdot {\hat{e}}_i$; in the reciprocal basis they are $V\cdot e_i$.