First let me define some notations.
For $n \geq 2$, the Cuntz Algebra $\mathcal{O}_n$ is the $C^*$-algebra generated by $n$ isometries $S_1, \dots, S_n$ satisfying $\sum_{i=1}^n S_iS_i^* = \textrm{id}$. Up to isomorphism, $\mathcal{O}_n$ does not depend on the choice of isometries. Denote $W_n$ to be the set of finite tuples $\mu = (\mu_1, \dots, \mu_k)$ for $k \in \mathbb{Z}$ with $\mu_m \in \{1, \dots, n\}$. Further, set $S_\mu = S_{\mu_1} \dots S_{\mu_k}$. Then define $\mathcal{D}_n$ to be the abelian $C^*$-subalgebra of $\mathcal{O}_n$ generated by the projections $S_\mu{S_\mu}^*$ for $\mu \in W_n$.
Now, in the literature it seems to be well-known that $\mathcal{D}_n$ is a MASA (maximum abelian subalgebra) of $\mathcal{O}_n$ and is referred to as the diagonal MASA. However, I could not find a proof for it. Even in the original paper by Cuntz and Krieger it is only mentioned that it is a MASA in a remark without proof. Could someone please point me to a reference which proves this or perhaps provide a proof?
Thank you.