In the classification of simple Lie algebras one has the familiar picture of 4 families, $A_n$, $B_n$, $C_n$ and $D_n$, and 5 exceptional groups, $F_4,$ $G_2,$ $E_6$, $E_7$ and $E_8$. The $D_n$ family has the unique feature that it contains, among all the corresponding Lie groups, groups whose center is non-cyclic: for $Spin(4n+2)$ the center is $\mathbb{Z}_4$, but one has $$Z(Spin(4n))=\mathbb{Z}_2\times \mathbb{Z}_2.$$ One can take the quotient of $Spin(4n)$ by any of the 3 subgroups of its center isomorphic to $\mathbb{Z}_2$ - one of these quotients is $SO(4n)$, the other are two are isomorphic to each other and are sometimes referred to as the *half-spin* or *semi-spin* groups, denoted by $SSpin(4n)$ (the notation may not be completely standard). They are a little bit the forgotten Lie groups - simple groups that are neither exceptional nor a quotient or covering of a classical group.

My question is: other than occuring in the classification, are there any places where these half-spin groups show up (naturally so to speak)?