Fix a Lie group $G$. The vector fields on $G$ form an (infinite-dimensional) Lie algebra with the commutator of vector fields as Lie bracket. **What (finite-dimensional) Lie algebras can I realise as subalgebras of this?**
(Let's suppose also that the vector fields must be non-vanishing, although I am happy to ignore this assumption if it helps).

I know that the set of left-invariant (or right-invariant) vector fields form a Lie algebra isomorphic to $\mathfrak{g}$, the Lie algebra of the group $G$. We can also get subalgebras of $\mathfrak{g}$ this way. Are there other Lie algebras (i.e. not isomorphic to $\mathfrak{g}$ or its subalgebras) that we can get by taking some appropriate subset of vector fields on $G$?

As a concrete example: Consider the group $G = SU(2)$. The left-invariant vector fields will form an $\mathfrak{su}(2)$ Lie algebra. Any single vector field on $SU(2)$ will form the abelian Lie algebra $\mathfrak{u}(1)$, and two commuting sets of vector fields will form the $\mathfrak{u}(1) \times \mathfrak{u}(1)$ Lie algebra. Can we find three vector fields which give us, say, the Heisenberg Lie algebra?

$[v_1,v_2]=v_3, \quad [v_2,v_3] = 0, \quad [v_1,v_3]=0$

If we were able to find three (globally defined) commuting vector fields, then these would form a three dimensional abelian Lie algebra $\mathfrak{u}(1) \times \mathfrak{u}(1) \times \mathfrak{u}(1)$, however there are topological restrictions to this, so it seems we can't simply obtain every finite dimensional Lie algebra.