What are the automorphisms of real Clifford algebras $Cl\_{n,0}$? Of course, I'm interested in the case where they are not central simple. 



Automorphisms of Clifford Algebras
$C\_{n,0}$ is either a full matrix algebra over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, or the direct sum of two such algebras that are isomorphic. The exact description depends on the residue class of $n$ (mod $8$) and can be found in textbooks or on wikipedia <http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras>


* $\mathrm{Mat}\_{k \times k}(\mathbb{R})$ and $\mathrm{Mat}\_{k \times k}(\mathbb{H})$ are central simple, so their automorphisms are all inner. (So the automorphism group is $\mathrm{PGL}\_k(\mathbb{R})$ or $\mathrm{PGL}\_k(\mathbb{H})$, respectively.)
* $\mathrm{Mat}\_{k \times k}(\mathbb{C})$ is not central simple over $\mathbb{R}$, so, in addition to the inner automorphisms, there is also complex conjugation. (So the group $\mathrm{PGL}\_k(\mathbb{C})$ of inner automorphisms is a subgroup of index $2$ in $\mathrm{Aut} \bigl( \mathrm{Mat}\_{k \times k}(\mathbb{C}) \bigr)$.)
* An automorphism of $A \oplus A$ can act independently on the two summands. Since $A$ is simple, the only additional automorphism interchanges the two summands. (So $\mathrm{Aut}(A) \times \mathrm{Aut}(A)$ is a subgroup of index $2$ in $\mathrm{Aut}(A \oplus A)$.)


