The Tits--Kantor--Koecher construction for Jordan dialgebras

We study a noncommutative generalization of Jordan algebras called Jordan dialgebras. These are algebras that satisfy the identities $[x_1{x}_2]x_3=0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1{x}_3)$, $x_1(x_1^2{x}_2)=x_1^2(x_1{x}_2)$; they are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. We present an analogue of the Tits—Kantor—Koecher construction for Jordan dialgebras that provides an embedding of such an algebra into Leibniz algebra.