Nilpotence of 
Commutator of 
The commutator of
was defined in 〚
Lie superalgebra structure on 〛.
In particular, when considering a commutator of an element of
and an element of
,
the following description is useful.
Consider the
– the universal enveloping algebra of
, and its
dual coalgebra
:
Consider the projector
:
which is identity on
and zero on all tensors of rank
containing at least one
.
This induces a map from
to
which we also
denote
:
For any Lie superalgebra
let
denote the commutator map:
In case of
, we can consider
as
a subspace in
using the projector
of Eq. (
45):
Then, the commutator on
satisfies:
Nilpotence of
We will now prove that
anticommutes with
:
When
,
by definition
. We must therefore check that
:
