Systems of linear equations and the word problem for the varieties \({\mathfrak N}_2{\mathfrak N}_c\) of Lie algebras (Q1307165)

Let \({\mathfrak N}_c\) be the variety of nilpotent Lie algebras of nilpotent class at most \(c+1\). The word problem for the varieties of polynilpotent Lie algebras \({\mathfrak N}_{c_1}{\mathfrak N}_{c_2}\cdots{\mathfrak N}_{c_k}\) was studied by many mathematicians. But the question of decidability for the word problem remained open for the varieties \({\mathfrak N}_2{\mathfrak N}_c\). Theorem 1. The word problem is decidable in the variety of Lie algebras \({\mathfrak N}_2{\mathfrak N}_c\). Corollary. The word problem is decidable in a variety \({\mathfrak M}\) of polynilpotent Lie algebras if and only if \({\mathfrak N}_3{\mathfrak N}_1\not\subseteq{\mathfrak M}\).