Relations between Clifford algebra and Dirac matrices in the presence of families

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of $gamma^a$s. Arranged into irreducible representations of eigenvectors of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= frac{i}{2} gamma^a gamma^b|_{a e b})$ these objects form $2^{frac{d}{2}-1}$ families with $2^{frac{d}{2}-1}$ family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by $tilde{S}^{ab}$ $=frac{1}{2} tilde{gamma}^a tilde{gamma}^b|_{a e b}$. Creation operators, carrying the family member and family quantum numbers form the basic vectors. The action of the operators $gamma^a$s, $S^{ab}$, $tilde{gamma}^a$s and $tilde{S}^{ab}$, applying on the basic vectors, manifests as matrices. In this paper the basic vectors in $d=(3+1)$ Clifford space are discussed, chosen in a way that the matrix representations of $gamma^a$ and of $S^{ab}$ coincide for each family quantum number, determined by $tilde{S}^{ab} $, with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding $d=(3+1)$ space into $d=(5+1)$-dimensional space.