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 $S^{ ab}$ $(= \frac{ i}{ 2} \gamma^a \gamma^b|_{ a \ne b})$ 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 $\tilde{ S}^{ ab}$ $=\frac{ 1}{ 2} \tilde{ \gamma}^a \tilde{ \gamma}^b|_{ a \ne b}$. 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.
Categories: physics.gen-ph hep-ph