Proper compact connected subgroup of $Spin(n)$ What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am only interested in the highest dimension of a compact connected subgroup of $Spin(n)$ of maximal rank. I am not sure if this is an easier question.
 A: I think that the answer here is just the double cover of the obvious answer for $SO(n)$, which is $U(n/2)$ when $n$ is even and $SO(n{-}1)$ when $n$ is odd.  You can double-check this by consulting the Dynkin tables of maximal subgroups.
Added after Mikhail's comment:  Mikhail actually went to the tables and checked (which I had not) and observed that, when $n$ is even, the maximal subgroup $SO(n{-}2)\times SO(2)$ of maximal rank has larger dimension than $U(n/2)$ when $n>8$.  (They have equal dimension when $n=8$ and the former has smaller dimension when $n<8$.)  Thus, the above answer needs to be divided into parts when $n$ is even.
By the way, the double covers of the subgroups $SO(6)\times SO(2)$ and $U(4)$ in $Spin(8)$ are actually conjugate by an outer automorphism of $Spin(8)$, so they are essentially the same.  This is a consequence of triality as discovered by Cartan.
A: A subgroup of maximal rank of maximal dimension is certainly a maximal subgroup of maximal rank.
Maximal connected subgroups of maximal rank in $Spin(n)$ correspond to maximal reductive Lie subalgebras of maximal rank in $so(n)_{\mathbf{C}}$.
Such subalgebras in semisimple Lie algebras were classified by Dynkin in 1952, see Onishchik and Vinberg (Eds.),
Lie Groups and Lie Algebras III, Encyclopaedia of Mathematical Sciences, vol. 41, Tables 5 and 6.
For $so(n)$ all such subalgebras are $so(2k)\oplus so(n-2k)$, and also $gl(n/2)$ for $n$ even.
The subalgebras of highest dimension are probably $so(n-1)$ for $n$ odd and $gl(n/2)$ for $n$ even.
EDIT: For $n=2l\ge 10$, the subalgebra of highest dimension and of maximal rank in $so(n)$ is $so(n-2)\oplus so(2)$ of dimension $2l^2-5l+4=l^2+l(l-5)+4$, and NOT $gl(n/2)$ of dimension $l^2$. For example, for $n=10$ we have
${\rm dim} (so(8)\oplus so(2))=29$, while ${\rm dim}\ gl(5)=25$.
