Finding the matrix representation of a geometric algebra It is well known that the matrix representation of $Cl_{3,0}(\mathbb{R})$ are the Pauli matrices. What is the general procedure to find the matrix representation of a geometric algebra?
If it helps, I am personally interested in finding the matrix representations of two specific cases (but would settle for a general approach):


*

*$Cl_{6}(\mathbb{C})$, with orthogonal generators $(\gamma_{\mu\nu}=\operatorname{diag}[1,-1,-1,-1,...])$:


$$
\gamma_{\mu\nu}=\frac{1}{2}(\gamma_\mu\gamma_\nu+ \gamma_\nu \gamma_\mu)
$$


*$Cl_{4}(\mathbb{C})$, but for for arbitrary generators (a.k.a the geometric algebra of general relativity):


$$
g_{\mu\nu}=\frac{1}{2}(e_\mu e_\nu + e_\nu e_\mu)
$$
 A: The Pauli matrices generate a representation of the even subalgebra of $Cl_{3,0}(\mathbb{R})$... the part of the algebra isomorphic to the quaternions.  To represent all of $Cl_{3,0}(\mathbb{R})$ you also need to represent the odd graded elements, which actually act like imaginary quaternions, with the grade-3 pseudo-scalar acting like a complex imaginary unit that commutes with the entire algebra.  The entire algebra is isomorphic to the complex quaternions (i.e. the biquaternions).
I know that there is an isomorphism between the 2x2 complex matrices and the complex quaternions.  Unfortunately, I don't know the general mechanism for finding the particular complex matrices that can serve the role of acting like each unit defined by the Clifford algebra, with the units squaring to positive or negative identity as appropriate while also maintaining the proper multiplication and commutation rules with each other such that the set of positive and negative basis elements is properly closed under multiplication.  In this case I would just guess that the pseudo-scalar/complex 'i' corresponds to the standard 90 degree rotation matrix with zeros on the diagonal and plus/minus one on the off diagonal, and then experiment with whether or not that plays nicely with the Pauli matrices.
I have a vague sense that representation theory provides some sort of concrete mechanism to enumerate the generators of an algebra with matrices of a given dimensionality, and that this mechanism is how the Pauli matrices were originally identified.  However I also find it entirely possible that we got the Pauli matrices via guess and check, so I'm not 100% certain that this algorithm exists.
This isn't a complete answer, but since your question is over a year old I'm hoping that by providing the context I am aware of, I am able to provide enough information to inspire a more complete proper answer.
