Commutators and anticommutators of the Pauli matrices

Let \(\sigma _ { i } , i = 1 , 2 , 3\) be the Pauli matrices.

1. Calculate \(\left[ \sigma _ { i } , \sigma _ { j } \right] \equiv \sigma _ { i } \sigma _ { j } - \sigma _ { j } \sigma _ { i } , i , j = 1 , 2 , 3 .\)

2. Calculate \(\{ \sigma _ { i } , \sigma _ { j } \} \equiv \sigma _ { i } \sigma _ { j } + \sigma _ { j } \sigma _ { i } , i , j = 1 , 2 , 3 .\)

3. Calculate \([ \sigma _ { i } , [ \sigma _ { j } , \sigma _ { k } ] ] + [ \sigma _ { j } , [ \sigma _ { k } , \sigma _ { i } ] ] + [ \sigma _ { k } , [ \sigma _ { i } , \sigma _ { j } ] ] .\)