Show that and three operators A cap B cap and C cap satisfy

Show that and three operators, A cap, B cap, and C cap satisfy: [A cap, B cap] = -[B cap, A cap] [A cap, B cap C cap] = B cap [A cap, C cap] + [A cap, B cap] C cap [A cap, B cap + C cap] = [A cap, B cap] + [A cap, C cap] (A cap + B cap) (A cap - B cap) = A cap^2 - B cap^2 if and only if [A cap, B cap] = 0

Solution

a). [A,B]=AB-BA

=-(BA-AB)

=-[B,A]

b) [A,BC] =B[A,C]+[A,B]C

rhs B[A,C]+[A,B]C

=B(AC-CA)+(AB-BA)C

BAC-BCA+ABC-BAC

=-BCA+ABC

ABC-BCA

=[A,BC]

C). [A,B+C] =A(B+C)-(B+C)A

=AB+AC-BA-CA

=(AB-BA)+(AC-CA)

. = [A,B]+[A,C]