Tameness of the cubic Chein automorphism

Determine whether the Chein automorphism of the free metabelian Lie algebra M_n of rank n\geq 4 given by (x_1+[[x_2,x_3],x_1],x_2,\ldots,x_n) is tame.

Background

The paper studies tame and almost tame automorphisms of the free metabelian Lie algebra M_n. It proves that every automorphism of M_n of rank n\geq 4 is almost tame, meaning that it is a product of Chein automorphisms, and further shows that the full automorphism group is generated by linear automorphisms together with specified quadratic and cubic automorphisms under the stated characteristic and rank conditions.

The displayed cubic Chein automorphism (x_1+[[x_2,x_3],x_1],x_2,\ldots,x_n) is identified as the first candidate for a wild automorphism in ranks n\geq 4. Resolving its tameness would determine whether the almost-tameness result can be strengthened to an exact analogue of the corresponding tameness theorem for free metabelian groups. The paper reduces the broader question of whether wild automorphisms exist to this specific automorphism, but does not resolve it.