Standard reference for a fundamental theorem on classification of root lattices

Question

In Schuett-Shioda's Mordell-Weil Lattices, the authors refer to a fundamental theorem on root lattices:

Theorem 2.25 Any positive-definite even integral root lattice is isometric to an orthogonal sum of root lattices of type $A_n$, $D_k$ and $E_\ell$.

Since they mention it so briefly, it seems like this is so well-known that barely needs mention.

But what would be a standard/classical reference for this theorem?

Answer 1

I saw this in Ebelings book "Lattices and Codes". In Chapter 1, the classfication into types $A_n, B_n$ and $D_n$ is proven.