Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.

We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.

For what type of Lia algebra structures $L$, $Gr(k,n)_{L}$ is a compact submanifold of ordinary Grassmannian $Gr(k,n)$?

Is there any relation between characteristic classes of canonical $k$- plane bundle on $Gr(k,n)_{L}$, and Lie algebraic invariants of $L$?