Consider a pair of matrices $(c_1, c_2)$. The words "it generates the 2-dimensional Lie algebra", means that there exists a pair of scalars $k_1$, $k_2$, such that $$[c_1, c_2] = k_1 c_1 + k_2 c_2,$$ where $[a,b]$ is the "commutator" $ab-ba$.
Not any pair generates 2-d Lie algebra.
Question: what is the dimension of the subset of matrices $(c_1,c_2)$ which generate 2-d Lie algebra? at least for $2\times 2$ matrices ?
It is clearly greater than $n^2+n$ since I can take - $c_1$ - arbitrary (so $n^2$) and $c_2$ - commuting with $c_1$, which gives ($+n$ for generic matrices). But in this way I get only $[c_1, c_2] =0$.
The motivation for the question comes from this question on MathOverflow.