The Brauer category $\mathcal{B}(2)$ has principal graph $D_{\mathrm{\infty }}$

We show that the subfactor planar algebra with principal graph $D_{\mathrm{\infty }}$ is the Brauer planar algebra with bubble constant $\delta =2$. The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At $\delta =2$, it relates to the category of representations of the orthogonal group $O(2)$. We work over any commutative ring with $1\mathrm{/}2$. Its principal graph encodes information about the corresponding monoidal category.