Lie automorphisms of the algebra of two generic \(2\times 2\) matrices (Q1975153)

Let \(R\) be the (associative) algebra generated by two generic \(2\times 2\) matrices \(x_1\) and \(x_2\) over a field \(K\) of characteristic 0. The authors study the Lie automorphisms of \(R\). These are the automorphisms of \(R\) that send \(x_1\) and \(x_2\) to linear combinations of commutators of \(x_1\) and \(x_2\). They prove that if \(\phi_{kl}(x_1)=x_1\), \(\phi_{kl}(x_2)=x_2+x_2\text{ ad}^{2k}x_1\text{ ad}^{2l+1}[x_1,x_2]\), \(k\geq 1\), \(l\geq 0\), then the subgroup of the Lie automorphisms of \(R\) generated by \(\text{GL}_2(K)\) and by \(\phi_{kl}\) is dense. Here the density is meant with respect to the formal power series topology.