  We invent a new cohomology theory for Lie triple algebras. Using this
cohomology, we introduce the notions of 2-term $L_\infty$-triple algebras and
Lie triple 2-algebras. We prove that the category of 2-term $L_\infty$-triple
algebras is equivalent to the category of Lie triple 2-algebras. Crossed
modules of Lie triple algebras are studied in detail.
