How do we embed a Spin group to an Orthogonal group?
Say a Spin($n$) group is a Lie group with $\frac{n \cdot (n-1)}{2}=$ Lie algebra generators.
Say a Spin(10) group is a Lie group with $\frac{10 \cdot 9}{2}=45$ Lie algebra generators.
Say an Orthogonal unitary group O($n$) has a $\frac{n \cdot (n-1)}{2}=$ Lie algebra generators.
Thus naively, we may attempt to
embed Spin($n$) into O($n$),
or Spin(10) into O(10).
questions:
What is the minimal $n$ to find the embedding of O($n$) $\supset \text{Spin}(10)?$
How do we find such an embedding by an explicit construction?
I suspect that O(16) $\supset \text{Spin}(10)?$ But what is the minimal $n$ allowed?
