Upper bound for isometric embeddings $\ell_2^m\rightarrow\ell_p^n$

The isometric embeddings $\ell_{2;\mathbb{K}}^m \rightarrow\ell_{p;\mathbb{K}}^n$ ($m\geq 2$, $p\in 2\mathbb{N}$) over a field $\mathbb{K}\in\lbrace \mathbb{R},\mathbb{C},\mathbb{H}\rbrace$ are considered, and an upper bound for the minimal $n$ is proved. In the commutative case ( $\mathbb{K}\neq\mathbb{H}$) the bound was obtained by Delbaen, Jarchow and Pełczyński (1998) in a different way.