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...γn−1}.

If x ∊ Γ then φ_{x}(1) = 1 and
φ_{x}(γ_{0}...γ_{n−1}) = γ_{0}...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.