I'm studying Knapp's book "Representation Theory of Semisimple Groups" and am trying to understand the structure theory of non-compact groups. Namely, let $G=KAN$ be the Iwasawa decomposition, $\mathfrak{g}$, $\mathfrak{k}$, $\mathfrak{a}$ be the respective Lie-algebras of $G$, $K$, $A$.
Define $\mathfrak{m}=Z_{\mathfrak{k}}(\mathfrak{a})$ the centraliser of $\mathfrak{a}$ in $\mathfrak{k}$. Then, on p.120, Knapp defines $\mathfrak{b}$ to be the maximal abelian subalgebra of $\mathfrak{m}$, thus, implying there may be non-commutative elements in $\mathfrak{m}$.
What is an example of a semi-simple Lie-algebra $\mathfrak{g}$ with the non-abelian $\mathfrak{m}$? For $\mathfrak{sl}(n)$, one should have $\mathfrak{m}=\mathfrak{b}=i\mathfrak{a}$, or at least I thought so.
EDIT: Appendix C in Knapp's book actually answers the question. All complex semi-simple groups have $\mathfrak{a}$ defined such that $\mathfrak{m} = i \mathfrak{a}$. For real Lie-groups, the split groups are those with $\mathfrak{m}=\mathfrak{b}=0$. The non-split groups such as $SL(n,\mathbb{H})$, $SU(p,q)$ have non-abelian $\mathfrak{m}$.