Simple Sagle algebras
A Sagle algebra over a field is a nonassociative algebra satisfying the identities \[ x^2=0,\quad J(x,y,z)w=J(w,z,xy)+J(w,y,zx)+J(w,x,yz) \] where \(J(x,y,z)=(xy)z+(yz)x+(zx)y\). Each Sagle algebra is a 3-Lie algebra in the sense of the author with respect to the triple multiplication \(\langle x,y,z\rangle =J(x,y,z)\). It is shown that if the characteristic of the base field is not equal to 3, then a non-Lie Sagle algebra is simple if and only if is it simple as a 3-Lie algebra. An arbitrary finite-dimensional simple Sagle algebra over a field of characteristic zero is a Lie algebra. The last result solves a problem raised by \textit{A. A. Sagle} in Pac. J. Math. 15, 1397-1400 (1965; Zbl 0241.17007).
scientific article; zbMATH DE number 1470958
scientific article