Angular Momentum II Using Bra and Ket Algebra

In Chapter 9 we had introduced the quantum mechanical angular momentum operator L and had shown that its components satisfy the following commutation relations: $$ \left. {\begin{array}{*{20}{c}}{\left[ {{L_x},{L_y}} \right] = i\hbar {L_z}} \\{\left[ {{L_y},{L_z}} \right] = i\hbar {L_x}} \\{\left[ {{L_z},{L_z}} \right] = i\hbar {L_y}} \\\end{array}} \right\} $$ (1) We had also shown that the square of the angular momentum operator L 2 (≡ L x 2 + L y 2 + L z 2 )commutes with L x , L y and L z ; i.e.