$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the compact subgroup of $GL(n,\mathbb{C})$ whose lie algebra is $\mathfrak{l}$, we have chosen a maximal commutative subalgebra $\mathfrak{t}$ of $\mathfrak{l}$ and we work with associated cartan subalgebra $\mathfrak{h}=\mathfrak{t}+i\mathfrak{t}$, we have chosen inner product on $\mathfrak{g}$ that is invariant under the adjoint action of $K$ and that takes real values on $\mathfrak{l}$, Consider the subgroups: $Z(\mathfrak{t})=\{A\in K: Ad_A(H)=H,\forall H\in \mathfrak{t}\}$ $N(\mathfrak{t})=\{Ad_A(H)\in \mathfrak{t},\forall H\in\mathfrak{t}\}$ I am not getting why $Z(\mathfrak{t})$ is a Normal subgroup of $N(\mathfrak{t})$, could any one help me to show this? and I must say I am not getting how the group look like I mean what type of element will be there in the group $W=N(\mathfrak{t})/Z(\mathfrak{t})$, please help in detail. thank you
this is a doubt from Brian C Hall page 174.