Ideally r-Constrained Graded Lie Subalgebras of Maximal Class Algebras

Let E ⊇ F be a field extension and M a graded Lie algebra of maximal class over E. We investigate the F -subalgebras L of M , generated by elements of degree 1 . We provide conditions for L being either ideally r -constrained or not just infinite. We show by an example that those conditions are tight. Furthermore, we determine the structure of L when the field extension E ⊇ F is finite. A class of ideally r -constrained Lie algebras which are not (r − 1)-constrained is explicitly constructed, for every r ≥ 1.