Solution
= Solution
Put $D=\mathbf a\cdot(\mathbf b\times\mathbf c)\ne0$. The <scalar triple product> is unchanged by cyclic permutation, so
$$
\mathbf A\cdot\mathbf a=\frac{\mathbf a\cdot(\mathbf b\times\mathbf c)}D=1,\quad\mathbf B\cdot\mathbf b=\frac{\mathbf b\cdot(\mathbf c\times\mathbf a)}D=1,\quad\mathbf C\cdot\mathbf c=\frac{\mathbf c\cdot(\mathbf a\times\mathbf b)}D=1.
$$
These are the diagonal pairings of a <reciprocal basis>.