Let $A$ be an associative $k$-Algebra. $W\subseteq A$ is said to be weakly closed if there is a $\gamma\colon W^2\to k$ such that $W$ is closed under the $γ$-bracket $[a,b]_γ=ab+γ(a,b)ba$.
Jacobson proves in his book about Lie algebras (Chap. II, Solvable and nilpotent lie algebras):
Theorem Let $W$ be a weakly closed subset of the Endomorphism algebra of a finite-dimensional vector space. If $W$ is nil (i.e., for every element $w\in W$, $w^k=0$ for some $k$), the enveloping associative algebra of $W$ is nilpotent.
The assumption makes it appear that the statement ceases to be true if we have an algebra other that $\mathrm{End}_k(V)$. What's an example of a nil $W$ weakly closed in $A$ such that the enveloping algebra is not nilpotent?