Is it true that given a Spin group $Spin(2N)$, the maximal special unitary subgroup that it can contain is $SU(N)$? So $$ Spin(2N) \supset SU(N)? $$
Is it true that given a Spin group $Spin(2N)$, the maximal unitary subgroup that it can contain is $U(N)$? $$ Spin(2N) \supset U(N)? $$ I think the 1. is true but the 2. is false. But how to prove or disprove them?
Because $(6)=(4)$, we may distinguish the case $N>4$, and $N<4$. I am mostly interested in the case $N>4$.
Some useful resources:
Pierre Deligne in his Notes on spinors