Every maximal subalgebra of rank ≤ n in W_n(K) is simple; rank-n submodule maximal subalgebras are ideal stabilizers and not simple; K[x,y]D is maximal for simple derivations D.
Matsumura,Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol
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Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$
Every maximal subalgebra of rank ≤ n in W_n(K) is simple; rank-n submodule maximal subalgebras are ideal stabilizers and not simple; K[x,y]D is maximal for simple derivations D.