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)?



Occurrence of semi-spin groups
Even if you are only interested in proving theorems about simply connected groups, you can naturally run into quotients like the half-spin groups. For example, you might try to prove something about your simply connected group by considering a reductive subgroup $G$ (maybe a centralizer of a rank 1 torus, for example). To prove the result for $G$, you may need to reduce to the case of the semisimple group $G/T$, where $T$ denotes the connected center of $G$ (a torus). Even when $G$ has derived subgroup $Spin(4n)$, the quotient $G/T$ could be $HSpin(4n)$, because $T$ can overlap the center of $Spin(4n)$.


Here is a specific example of this that some people care about, where $HSpin(12)$ arises from $E\_7$. (This can be viewed as a follow-on to Christian's answer--this example is less "natural" than his.) One of the very few remaining cases of the Kneser-Tits Conjecture [[google](http://www.google.com/search?q=kneser-tits+conjecture)] concerns a simply connected group of type E7 with rank 1--deleting the distinguished vertex leaves a subdiagram of type $D\_6$. If you attempt to prove rationality of the $E\_7$ by standard reductions as in [Chernousov-Platonov](http://www.ams.org/mathscinet-getitem?mr=1656830), then you end up doing the procedure I outlined in the previous paragraph, and you are stuck with proving rationality for a form of $HSpin(12)$.


