The $Spin(p,q)$ group is usually defined as $$ Spin(p,q) = \{s \in C\ell^{+}_{p,q}: s \tilde{s}=1\} $$ where $s$ belongs to the Lipschitz group and $C\ell^{+}_{p,q}$ denotes the even element subset of the Clifford algebra with signature $(p,q)$. Since in this definition the Clifford algebra is defined over the reals, the tilde in $\tilde{s}$ simply refers to inverting the order of the elements inside $s$.
I have two questions:
- Does this definition for the tilde still work for the Spin group definition if I consider instead the complexified Clifford algebra? I.e., is this definition correct: $$ Spin(p,q) = \{s \in \mathbb{C}\otimes C\ell^{+}_{p,q}: s \tilde{s}=1\} \, ? $$ My intuition says that $\tilde{s}$ should be redefined as the elements in inverse order followed by a complex conjugation of the coefficients contained in it, i.e. it should actually appear as $\tilde{s}^*$ in the definition of the Spin group, by I'm not sure of this.
- Is there still a Triality automorphim in the case of the $Spin(8,0)$ group considering complex coefficients?
Thanks!