Centralizer of diagonal matrices in $SL_n(\mathbb{R})$

Question

Let $D$ be the subgroup of $SL_n(\mathbb{R})$ containing all the diagonal matrices (of $SL_n(\mathbb{R})$). I want to find the centralizer of $D$ in $SL_n(\mathbb{R})$, namely $C_{SL_n(\mathbb{R})}(D) = \{s \in SL_n(\mathbb{R})| sd = ds \hspace{2.5mm} \forall d \in D \}$. My guess is that it is $D$ itself, since only diagonal matrices commute with all diagonal matrices. Am I missing something here?