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 odd integer $2n+1$: $$ Spin(2n+1) \subseteq SU(2^{n}). \tag{1} $$ Here $Spin(2n+1):= Spin(2n+1,\mathbb{R})$, while the special unitary group $SU(2^{n })\subset SL(2^{n },\mathbb{C})$.
I would like to know whether eq.(1) can still be true in general.
For example,
$n=1$, it is true $Spin(3)=SU(2)$.
$n=2$, it is true $Spin(5) \subseteq Spin(6) =SU(4)$.
$n=3$, it is $Spin(7) \subseteq SU(8)$? likely true.
Whether this formula eq.(1) is true for any $n \geq 1$? or are there loopholes? like this loophole case here $Spin(2n) \subseteq SU(2^{n-1})$ for $n \geq 5$??
