Nilpotent Lie algebras in which all proper subalgebras have class at most n

Let L be a finitely generated nilpotent Lie algebra over a field K and let d be the smallest integer such that L can be generated by d elements. Let n ≥ d be a positive integer and suppose that every proper subalgebra of L has class at most n. It is not difficult to show that the class of L is at most n + q where q = ⌊ n / ( d − 1 ) ⌋ . Our main result shows that there exist such Lie algebras of class (exactly) n + q whenever q ≥ 3 and K has characteristic 0 or prime characteristic p such that p does not divide ( q 2 − 1 ) q / 2 .