Is the trace of the product of a Hermitian and a Pauli vector zero?  No.  Ends up kind of dot product like.

(*$Assumptions = *)

h = {{a,b-I c},{b + I c, -d}};
sigmadot[x_,y_, z_] := PauliMatrix[1] x +  PauliMatrix[2] y  + PauliMatrix[3] z;

(h.sigmadot[e,f,g] // Tr) // Simplify