On almost Engel \(L\)-varieties of vector spaces (Q2145067)

Let \(A\) be an associative algebra over a field \(F\) and \(V\) an \(F\)-subspace of \(A\) generating \(A\) as an algebra. An element \(f(x_1,\dots,x_n)\) of the free associative algebra is an \emph{identity of the pair} \((A,V)\) if \(f(v_1,\dots,v_n)=0\in A\) for all \(v_1,\dots,v_n\in V\). An \emph{\(L\)-variety} consists of all pairs that satisfy a given set of identities, and an \(L\)-variety is \emph{almost Engel} if it does not satisfy an Engel identity, but all its proper \(L\)-subvarieties satisfy an Engel identity. In the present paper it is proved that there are exactly two almost Engel \(L\)-varieties generated by a pair of the form \((A,A)\), namely the varieties generated by \((A_1,A_1)\) and \((A_2,A_2)\), where \(A_1=\{\left(\begin{array}{cc}a & b \\ 0 & 0\end{array}\right)\mid a,b\in F\}\) and \(A_2=\{\left(\begin{array}{cc}a & 0 \\ b & 0\end{array}\right)\mid a,b\in F\}\).