My question is simple: I have an element of Spin(4) written in terms of $16\times 16$ matrices from a Clifford algebra $C\ell_{8,0}$. Does this imply that any such element is also an element of Spin(8)?
I understand that the other way around doesn't need to hold since there are things built in a 8-dimensional space that cannot be built using solely quantities from a 4-dimensional space. However, I'm not sure if every element of Spin(4) belongs to Spin(8) as well.
Any help is greatly appreciated!