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 $Gr(k,n)$?

Can the Lie algebra structure of $L$ be determined by topology of $Gr(k,n)_{L}$?That is: Are there two non isomorphic Lie algebra structures $L$ and $L'$ on $\mathbb{R}^{n}$ such that $Gr(k,n)_{L}$ is homemorphic to $Gr(k,n)_{L'}$?

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