An explicit construction of a limit variety of groups

We construct an explicit countable group $G_{\ast }=N(R)⋊A$ whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here $A$ is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and $R$ is an explicitly defined quotient of the group algebra ${\mathbb{F}}_2[A]$. The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in ${\mathfrak{B}}_4{\mathfrak{B}}_2$. This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.