Reposted from physics.SE. Hope the formulation of the question will be clear as it is. Do let me know in case I should be more clear or precise!

Suppose that we have a physical system with some symmetry, given by a Lie group whose generators are $Q\_i$. Suppose that there is a (linearly independent) subset of these generators, $Q\_A$, which commute with one another, and in general satisfy the relation
$$
[Q\_i,Q\_A]=(T\_i)^B\_{\phantom BA}Q\_B,
$$
where the matrices $T\_i$ define a representation of the Lie algebra of $Q\_i$. Is it always possible to find a basis of the Lie algebra of the form $\{Q\_a,Q\_A\}$, where the generators $Q\_a$ themselves form a closed subalgebra?

An example of such a structure would be the group $\text{ISO}(n)$ of isometries of the $n$-dimensional Euclidean space. Out of its generators, the $n$ translations form an Abelian ideal (my $Q\_A$), whereas the $n(n-1)/2$ rotations form a non-Abelian subalgebra (my $Q\_a$). The whole group $\text{ISO}(n)$ is a semidirect product of the subgroups of rotations and translations.

I believe that the answer to my question is in general *no*, but I would love to see some (possibly simple) examples.