Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices.
The parabolic subgroups $P$ containing $B$ have $1-1$ correspondence to the subsets $\Pi_s$ of the set of simple roots $\Pi$. As an example, Consider $$ P=\left\{\begin{pmatrix}*&*&*&*&*&*\\*&*&*&*&*&*\\&&*&*&*&*\\&&*&*&*&*\\&&*&*&*&*\\ &&&&&* \end{pmatrix}\right\} $$ It is a parabolic subgroup corresponds to $\Pi_s=\{\epsilon_1-\epsilon_2,\epsilon_3-\epsilon_4,\epsilon_4-\epsilon_5\}$, where $\epsilon_j(\mathrm{diag}(a_1,\cdots a_n))=a_j$. The parabolic subalgebra of $P$ has Langlands decomposition $$ \mathfrak{q}=\mathfrak{m}\oplus\mathfrak{a}\oplus\mathfrak{n} $$ with $$ \mathfrak{a}=\left\{\mathrm{diag}(a,a,b,b,b,c), 2a+3b+c=0\right\}. $$ The Weyl group $$ W(G,A):=N_K(\mathfrak{a})/C_K(\mathfrak{a}), $$ where $K=SO(n)$ the associated maximal compact subgroup of $G$.
How to calculate $C_K(\mathfrak{a})$ and $N_K(\mathfrak{a})$? Is $W(G,A)=\{I\}$ in the above example?
In this case the Weyl group is trivial. For $SL(n,\Bbb R)$, if we set $A_o$ equal to the usual diagonal split component of $B$ then the Weyl group of $A_o$ is $S_n$ acting by permuting the diagonal components. The Weyl group of $A$ is the set of elements of the Weyl group of $A_o$ that preserve $A$ the standard split component of the parabolic block upper triangular subgroup $P$. In the example, if a permutation preserves the set of diagonal elements of the form $(a,a,b,b,b,c)$ then it must be in $S_2\times S_3\times S_1$ acting in the obvious way on $6$-tuples hence it acts trivially on A.
In general, the centralizer of a split component of a parabolic is the corresponding Levi factor, say $M$. The normalizer is gotten as the union of the $kM$ with $k$ a representative in the chosen maximal compact subgroup (in this case,$SO(n)$) that preserve the subgroup $A$ of $A_o$. For $SL(n,\Bbb R)$ the representatives for $A_o$ can be taken to be the permutation matrices with appropriate minus signs to make the determinant equal to $1$.