Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric spaces" defines the Weyl group $W(G,K)$ as follows.

Let $\mathfrak{g}=Lie(G), \mathfrak{k}=Lie(K)$. Let $$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$ be the Cartan decomposition. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$. Let $M$ and $M'$ be respectively the centralizer and the normalizer of $\mathfrak{a}$ in $K$. Clearly $M$ is a normal subgroup of $M'$. Then $$W(G,K):=M'/M$$ is called the Weyl group of $G/K$ and is known to be finite.

**Question.** Is there a direct relation between $W(G,K)$ and the Weyl group $W$ of $G$ (or may be better to say of the compact form of $G$)?
E.g. can one say that $W(G,K)\subset W$ in some natural way?

A reference would be very helpful.

Lie Groups Beyond an Introduction(Birkhauser, 1996). There are other books, for example an older one by Vogan. Keep in mind that Helgason is focused especially on symmetric spaces rather than just Lie groups and their Lie algebras; but your question involves mostly the latter. Helgason also leaves some of the pieces like this one for the reader to assemble, so it's well worth consulting other sources too. $\endgroup$