When is a subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^N$ isometrically isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}^n$?

It is shown in this note that one can decide whether an $n$-dimensional subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^N$ is isometrically isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an $n$-dimensional subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^N$ with projection constant equal to one must be isometrically isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}^n$.