Algorithm for constructing 
We can construct such subalgebras in the following way.
Start with some subspace
satisfying the property:
with some
; this implies that
.
Verify the existence of
such that:
If this is not the case, then we must declare failure (the original space
was a bad choice). But if Eq. (
36) is satisfied, then we define:
It is useful to identify the subspace in the tensor algebra of
consisting of tensors with the symmetries of the free Lie algebra.
For example, we may interpret
as degree two elements of the free Lie algebra generated by
, because
at the degree two the only relation in the free Lie algebra is antisymmetry:
(where
stands for “the commutator in the free Lie algebra”).
We observe that under the condition (
36) the “nested commutator”
(with
,
and
being any three elements of
) satisfies the Jacobi
identity:
Verify the existence of
such that:
where
is the commutator in the free Lie algebra.
If this is true, define:
Otherwize, say that the original space
was a bad choice.
Continue recursively. For any Lie algebra, let us denote the “nested commutator”
as follows:
At each step, the only thing that we have to verify the existence of
such that:
This automatically implies, that any expression composed of
letters
(elements of
) using the operation
(
e.g. the nested commutator
(
38)) automatically satisfies all the relations of the free Lie algebra.
In order to prove this, we notice that the
nested commutator
, provided that the condition (
39) holds
for any
, is obtained from
by putting
in front of
out of
es, does not matter which ones.
We only need to verify (
39), because other Poisson brackets will be
-exact
automatically; for example:
We define 
as the space of expressions composed from elements
of using .
Finally:
If this construction works (
i.e. if we are able to verify the existence
of
satisfying Eq. (
39) at every step), then the resulting
can be understood as the space generated by the expressions obtained
from generators of
by using operations
and
, so that
all but one generators
enter as
(and one remaining without
). Notice that all the expressions where the number of
’s is less than
the number of
minus one are automatically
-exact, therefore Eq. (
25).
Another interpretation (or rather consequence) of Eq. (
39) is that those expressions
constructed of from
and commutators, satisfy all the relations of the free Lie
algebra, for example:
There are generally speaking other relations between them, and therefore they are
not a free Lie algebra. But they are
a Lie algebra. We will denote this Lie algebra
.
with some
It is useful to identify the subspace in the tensor algebra of
(where
Otherwize, say that the original space
by all the possible symmetries
,...,
and thus “filling the multiplet”.
