A Note on a Result of Kostrikin

On a class of finitary Lie algebras characterized through derivations

“…It does not seem completely obvious how to proceed to larger integers n. However, an adaptation of the arguments from [9] works. First we notice that (1) implies that…”

“…A glance at the proof of [9, Lemma 2.2] then shows that this lemma holds for d and R a (resp. L b ) playing the role of X and A (here we have referred to the notation in [9]). Next, following the strategy in [9] we note that (DA)D ⊆ D will be established by proving that for each pair a, b ∈ A there exists ξ ∈ End(A) such that…”

“…Fortunately, this does not affect the proof. One just has to follow literally all steps from [9] to arrive at the desired conclusion,…”

“…The proof combines the recently developed structure theory of finitary Lie algebras and Lie algebras with minimal inner ideals with the results on derivations in general nonassociative algebras, obtained in Section 2. The basic result from that section is hidden in the arguments from the recent paper by E. García and M. Gómez Lozano [9].…”

A radical for graded Lie algebras

“…In particular x g is an ad-nilpotent element of L Φ , and therefore an ad-nilpotent element of L of index n, i.e., there exists n ∈ N such that ad n xg (L) = 0 but ad n−1 xg (L) = 0. By [5] let 0 = y ∈ ad n−1 xg (L) be a homogeneous ad-nilpotent element of index 3 and let us consider the G-graded Jordan algebra L y . In L y every element is nilpotent of bounded index since y is still strongly Engel in L and we can argue as in [6,Proposition 3.2].…”