Maximal commutative subalgebras A of M_n(K) satisfy dim A ≥ n for n ≤ 13, Courter's n=14 example is optimal, and stack constructions attain the bound for all n ≥ 14.
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For n=6, every maximal commutative subalgebra of M_6(k) has dimension at least 6.
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Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds
Maximal commutative subalgebras A of M_n(K) satisfy dim A ≥ n for n ≤ 13, Courter's n=14 example is optimal, and stack constructions attain the bound for all n ≥ 14.
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On the minimal dimension of maximal commutative subalgebras of $M_6(k)$
For n=6, every maximal commutative subalgebra of M_6(k) has dimension at least 6.