I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the following paper:


Yanovski, A. B. "Linear bundles of Lie algebras and their applications." Journal of Mathematical Physics 41.11 (2000): 7869-7882.


But surely this is a much older result, at least for the case of matrix algebras.



Earliest source for a Lie algebra construction
I think it has first been considered by A.A. Albert in $1948$, in connection with
so-called [*Lie-admissible algebras*](http://en.wikipedia.org/wiki/Lie-admissible_algebra). An algebra $(A,\cdot)$ is called Lie-admissible, if $[a,b]=a\cdot b-b\cdot a$ defines a Lie bracket on the vector space of $A$. The bracket $[a,b]=axb-bxa$ has been considered later (around $1967$ for the bracket $[a,b]=\lambda ab-\mu ba$ for scalars $\lambda,\mu$, and $1978$ for the bracket $[a,b]=apb-bqa$ for fixed $p,q$ in an associative algebra) in physics (Lie-Santilli bracket). For references see (for example) [here](http://arxiv.org/pdf/math/0609501), page $8$.


