Let $V$ be a dimension $n>0$ vector space over $\mathbb{C}$ and let $$ V=V_1 \supset V_2 \supset \dots \supset V_r \supset V_{r+1} = \{0\} $$ be a (not necessarily complete) flag of subspaces.
Is there any description in the literature for the space $$ \{ \phi \in \mbox{End}(V) \mid \phi(V_i) \subset V_i, \, 1\leq i \leq r \} $$ of endomorphisms of $V$ that preserve the flag?
I'm searching for references that deal with its Lie algebra structure and matrix description. I'm mostly interested in the particular case if we replace $\mbox{End}(V)$ by $\mathfrak{sl}(V)$.
References for particular cases are wellcome as well.