Ideally $r$-constrained graded Lie subalgebras of maximal class algebras

Abstract

Let $E\supseteq F$ 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 $E\supseteq F$ is finite. A class of ideally $r$-constrained Lie algebras which are not $(r-1)$-constrained is explicitly constructed, for every $r\geq 1$.