CommutatorExplicit[exp] substitutes any
Commutator and AntiCommutator in
exp by their definitions.
DeclareNonCommutative[a, b, c, d]Commutator[a, b]
CommutatorExplicit[%][a,b]
a.b-b.a
AntiCommutator[a - c, b - d]
CommutatorExplicit[%]\{a-c,\medspace b-d\}
(a-c).(b-d)+(b-d).(a-c)
CommutatorExplicit[AntiCommutator[a - c, b - d]] // DotSimplifya.b+b.a-a.d-d.a-b.c-c.b+c.d+d.c
UnDeclareNonCommutative[a, b, c, d]