Calculate the product of two vectors in their Pauli (sigma) matrix representation, and then take the trace of that.

(PauliMatrix[1].PauliMatrix[2]) // MatrixForm
( PauliMatrix[2]. PauliMatrix[3]) // MatrixForm
 (PauliMatrix[3]. PauliMatrix[1])// MatrixForm

(PauliMatrix[1]. PauliMatrix[2]. PauliMatrix[3])// MatrixForm

(PauliMatrix[1].PauliMatrix[2]) // Tr
( PauliMatrix[2]. PauliMatrix[3]) // Tr
 (PauliMatrix[3]. PauliMatrix[1])// Tr

(PauliMatrix[1]. PauliMatrix[2]. PauliMatrix[3])// Tr

ClearAll[a, b,c, d, e,f ]
x = a PauliMatrix[1] + b  PauliMatrix[2] + c PauliMatrix[3];
x// MatrixForm
y = d PauliMatrix[1] + e  PauliMatrix[2] + f PauliMatrix[3];
y // MatrixForm
z = x.y;
z// MatrixForm
(z// Tr)//Simplify