  The purpose of this paper is to study the construction of $3$-Bihom-Lie
algebras. We give some ways of constructing $3$-Bihom-Lie algebras from
$3$-Bihom-Lie algebras and $3$-totally Bihom-associative algebras. Furthermore,
we introduce $T_\theta$-extensions and $T_\theta^*$-extensions of $3$-Bihom-Lie
algebras and prove the necessary and sufficient conditions for a
$2n$-dimensional quadratic $3$-Bihom-Lie algebra to be isomorphic to a
$T_\theta^*$-extension.

Rings and Algebras