Automorphisms of Clifford Algebras

Let C be the Clifford algebra generated by the vector space V. An automorphism of C is ipso-facto an automorphism of V along with its quadratic form. The automorphisms of a vector space with quadratic form are just those linear transformations that preserve the quadratic form — i.e. the orthogonal transformations O(n). If φ is an automorphism on C then φ restricted to V is an orthogonal transformation on V which is C’s generator. α(v) (which changes the signs of the odd grades of C) is an automorphism on a Clifford algebra. If x∊V then α(x) = −x.

If x is a member of the Clifford group (Γ) then φ_x = λz.xzα(x)⁻¹ is an inner automorphism. For all z∊C, φ_x(z) = xzα(x)⁻¹.

In Cl(n):
α = φ_{γ₀...γ_n−1}.
If x ∊ Γ then φ_x(1) = 1 and φ_x(γ₀...γ_n−1) = γ₀...γ_n−1.

This does not address the question of whether another subspace of a Clifford algebra could serve in place of V. A few minutes of thought found no automorphisms that failed to map V to itself. The ‘subspace of the reals’ must be mapped to itself by any automorphism.