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.



Weyl group of a symmetric space
The answer is in the book you quote (1978 version) on page 325.


