The problem is to determine when a smooth,
k
k
-dimensional distribution
D
k
{D^k}
defined on an
n
n
-manifold
M
n
{M^n}
, locally admits a vector field basis which generates a nilpotent, solvable or even finite-dimensional Lie algebra. We show that for every
2
≤
k
≤
n
−
1
2 \leq k \leq n - 1
there exists a (nonregular at
p
∈
M
n
p \in {M^n}
) distribution
D
k
{D^k}
on
M
n
{M^n}
which does not locally (near
p
p
) admit a vector field basis generating a solvable Lie algebra. From classical results on the equivalence problem, it is shown that for
1
≤
k
≤
4
1 \leq k \leq 4
and
D
k
{D^k}
regular at
p
∈
M
4
p \in {M^4}
,
D
k
{D^k}
admits a local vector field basis generating a nilpotent Lie algebra.