According to this post Embed a Spin group to a special unitary group
It seems to me that there is an answer proposes that for even integer $2n$: $$ Spin(2n) \subseteq SU(2^{n-1}). \tag{1} $$ Here $Spin(2n):= Spin(2n,\mathbb{R})$, while the special unitary group $SU(2^{n-1})\subset SL(2^{n-1},\mathbb{C})$.
For some reason given here https://mathoverflow.net/q/295711/336737, the $n=4$ answer seems to be false. Namely, $Spin(8) \not \subset SU(8)$. I would like to know whether eq.(1) can still be true in other $n$ or in general.
For example,
$n=1$, it is false.
$n=2$, it is false.
$n=3$, it is $Spin(6)=SU(4)$ true.
$n=4$, it is $Spin(8) \not \subset SU(8)$ here https://mathoverflow.net/q/295711/336737, so false.
Whether this formula eq.(1) is true for $n \geq 5$?
