  We study an extension algebra $A$ from two given $3$-Lie algebras $M$ and
$H$, and discuss the extensibility of a pair of derivations, one from the
derivation algebra of $M$ and the other from that of $H$, to a derivation of
$A$. In particular, we give conditions for such an extension to be a $3$-Lie
algebra, and provide necessary and sufficient conditions of the pair of
derivations to be extendable.

Rings and Algebras