Classification of certain types of maximal matrix subalgebras

Let Mn(K) denote the algebra of n×n matrices over a field K of characteristic zero. A nonunital subalgebra N⊂Mn(K) will be called a nonunital intersection if N is the intersection of two unital subalgebras of Mn(K). Appealing to recent work of Agore, we show that for n≥3, the dimension (over K) of a nonunital intersection is at most (n−1)(n−2), and we completely classify the nonunital intersections of maximum dimension (n−1)(n−2). We also classify the unital subalgebras of maximum dimension properly contained in a parabolic subalgebra of maximum dimension in Mn(K).