 Since superoperators are themselves operators the use of the super- prefix is used to distinguish them from the operators upon which they act.Defining the left and right multiplication superoperators by formula_1 and formula_2 respectively one can express the commutator as

formula_3 

Next we vectorize the matrix formula_4 which is the mapping

formula_5

The matrix representation of formula_6 is then calculated by using the same mapping

formula_7

indicating that formula_8.Similarly one can show that formula_9.