When does an frame of rank $n$ contain $n$ orthonormal vectors?

Question

Fritz John's ellipsoid theorem gives the minimal ball containing a given convex body $K$. Moreover, we have for $m$ points in boundary
$$(A) \quad \sum_m c_iu_iu_i^{T} = I_n,$$
where $I_n$ is the identity matrix, $c_i$ some constants, and $u_i$ vectors representing points in $\partial(K) \cap \partial(B_n(0,1)).$ Now, I would like to see conditions that quarantee that $u_i$ where $i=1,\dots m$ contain $n$ orthonormal vectors. For example (A) seems to me a lot like a projection in $m$ dimension of rank $n.$ I would be interested of canonical forms of this projection. On the other direction the vectors $u_i$ seem to me to constitute a frame. So I am asking conditions when there is an orthonormal subset of $n$ vectors of $\{u_i\}.$ I would have good uses for them. Now, that $u_i \in \partial(K) \cap \partial(B_n(0,1))$ and that they are orthonormal implies that $K$ contains a standard $n$-simplex. The entries of the covariance matrix of a convex body $K$ are defined as \begin{equation} \label{last} (a_{ij}) = \frac{\int_K x_ix_j}{|K|} - \frac{\int_K x_i}{|K|}\frac{\int_K x_j}{|K|}. \end{equation} We define the isotropic constant of any convex body $K$ in scaling invariant way using \begin{equation} \label{last2} L^{2n}_{K} :=\frac{\text{Det}\text({Cov{K}})}{|K|^2}. \end{equation}
Now, if I assume that the minimal ellipsoid containing the convex body $K$ is in the unit ball and the body is isotropic, I have (B) $$L^2_K = \int_K \frac{x_ix_i}{|K|^{1+2/n}} \leq |K|^{-2/n}.$$ But if the simplex $S_n \subset K,$ then I have $$L^2_K \leq |K|^{-2/n} \leq |S|^{-2/n} = (n!)^{2/n}.$$ (EDIT)(EDIT) For the following I don't even need the set inclusion $S_n \subset K$ because if the maximal ellipsoid is the unit ball, a well known volume-ratio estimate shows that $|S_n| \leq |K|,$ with $S_n$ the unique minimizer. Moreover, if I have estimate $$m_{n-1}(e^{\perp} \cap K) < m_{n-1}(e^{\perp} \cap S_n),$$ this would lead to a contradiction if the slice of $K$ is in a John's position. This is because the slice of the simplex is a lower dimensional simplex. This should mean that the simplex is the extremal case. (EDIT) However, it seems that I don't always have the minimal ellipsoid to be a ball after I take the intersection $K \cap S_n.$ So I don't always obtain a frame with $n$ orthogonal vectors.