Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$.
For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such that $J^2=-I$ are matrix representations of $\mathbb{C}$.
This means that any matrix of the form
$$
J=\left[
\begin {array}{cccc}
a&b\\
c&-a
\end{array}
\right]
\qquad bc \le-1 \qquad a=\sqrt{-1-bc}
$$
is a possible representation of the imaginary unity $i$ and a complex number $z=x+iy$ is represented by the matrix $M(z)=Z=xI+yJ$ I think that this fact can be see as a consequence of the Artin–Wedderburn theorem about the matrix representation of rings ( see my answer to: What does it mean to represent a number in term of a $2\times2$ matrix?),
that generalize such kind of representation to all semisimple rings.

Now, consider $\mathbb{C}$ as a $^*$Algebra (with complex conjugation as the involution). If we search a $^*$Algebra isomorphism such that $M(\bar z)=M^T(z)$ (where the involution for the matrix ring is the transpose), we see that there are only two possible representations, with :

$$
J=\left[
\begin {array}{cccc}
0&1\\
-1&0
\end{array}
\right]
\qquad \mbox{or}\qquad
J=\left[
\begin {array}{cccc}
0&-1\\
1&0
\end{array}
\right]
$$
So we have a great restriction of the possible representations.

My question is if this restriction is a consequence of some general representation theorem for *Algebras in the same sense as the matrix representation of the field $\mathbb{C}$ is a consequence of the A-W theorem.

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