I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.

One can show that $A \in \mathbb{R}^{m \times n}$ is an isometry from $\ell_p^n(\mathbb{R}^n) \to \ell_p^n(\mathbb{R}^m)$ if and only if the columns of $A$ have unit $\ell_p$-norm and only one nonzero entry per column. Taking $n \le m$, this characterises the subspaces isometric to the subspace $\ell_p^n(\mathbb{R}^n) \subset \ell_p^n(\mathbb{R}^m)$.

Also, for $p$ not an even integer, I believe all the isometries between subspaces of $L_p([0,1])$ have been classified, and this can likely be extended to $\ell_p^n(\mathbb{R})$.

Does anyone know any references in this direction for even $p$?