I need to use the following theorem:

Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of $\mathfrak{g}$. Then any restricted weight $\lambda$ of $\rho$ satisfies $$ \forall \alpha \in \Sigma,\quad 2\frac{\langle \lambda, \alpha \rangle}{\langle \alpha, \alpha \rangle} \in \mathbb{Z}. $$

The case of non-restricted weight and roots is of course well-known and easily found in the literature. For the general result, the closest things I found are Propositions II.4.21 to II.4.23 in S. Helgason, Geometric Analysis on Symmetric Spaces; and the last page of the proof of Theorem 8.49 in A.W. Knapp, Lie Groups Beyond an Introduction. But neither of these two passages gives quite exactly the result I am looking for.

I am not asking for a proof: I already have one (it is not very hard). But it looks like such a basic thing that it should already be written somewhere. If anyone knows any reference it would be very much appreciated!