We introduce the conception of matched pairs of $(H, \beta)$-Lie algebras, and construct an $(H, \beta)$-Lie algebra through them. We prove that the cocycle twist of a~matched pair of $(H, \beta)$-Lie algebras can also be matched.
"Matched pairs of generalized Lie algebras and cocycle twists." J. Gen. Lie Theory Appl. 3 (2) 149 - 155, May 2009. https://doi.org/10.4303/jglta/S090205