John's ellipsoid theorem characterizes the maximal-volume ellipsoid contained in a given convex body by an algebraic condition on the contact points. A classical consequence of this result is the upper bound by $n$ on the Banach–Mazur distance between an arbitrary convex body and the Euclidean ball; in the centrally symmetric case, this bound improves to $\sqrt n$. In this talk, we present an analogous algebraic characterization on the contact points, but directly for the ellipsoid attaining the Banach–Mazur distance. Our proof is based on an extension of an idea originating in a 1938 paper by O. B. Ader, a student of Fritz John. This approach allows us to rederive the classical upper bounds $n$ and $\sqrt n$ without appealing to volume arguments. Moreover, our method yields precise information about the cases of equality in these bounds, filling several gaps in the existing understanding of the Banach–Mazur distance to the Euclidean ball. In particular, we obtain a simple proof that in dimension $n=3$ the cube and the cross-polytope are the only centrally symmetric convex bodies with maximal Banach–Mazur distance to the Euclidean ball. We also show that our characterization leads to a quick proof of Maurey's theorem on the uniqueness of the Banach–Mazur ellipsoid, a result that has not previously appeared in the literature. Finally, we investigate the Banach–Mazur position of two arbitrary convex bodies. In this general setting, the algebraic characterization of contact points remains a necessary condition, but is no longer sufficient in general.