On the inclusion ${\text{cO}}_2\subset {\text{cQ}}_2$

The diadic $C^{\ast }$-algebra ${\text{cQ}}_2$ contains canonically a copy of the Cuntz algebra ${\text{cO}}_2$. It is shown that the inclusion ${\text{cO}}_2\subset {\text{cQ}}_2$ is $C^{\ast }$-irreducible and rigid. It follows that the injective envelopes of these two $C^{\ast }$-algebras are $\ast$-isomorphic.