The required isomorphism between $*$-rings, with a direct and constructive proof, is stated and proved [1]. The required theorem is given as Theorem 2.

[1] - "Matrix representations of Clifford algebras". Linear Algebra and its Applications. 1990.

The required isomorphism between $*$-rings, with a direct and constructive proof, is given here [1]. The required theorem is given as Theorem 2. The proof is all of section 3.

[1] - "Matrix representations of Clifford algebras". Linear Algebra and its Applications. 1990.