Conjugacy Cartan subalgebras in \(n\)-Lie algebras (Q1918790)

\(n\)-Lie algebras were introduced by \textit{V. T. Filippov} [Sib. Math. J. 26, 879-891 (1985); translation from Sib. Mat. Zh. 26, No. 6, 126-140 (1985; Zbl 0585.17002)]. An \(n\)-Lie algebra \(A\) over a field \(k\) is a vector space endowed with an \(n\)-ary skew linear multiplication \([x_1, \dots, x_n]\) such that \[ \bigl[[x_1, \dots, x_n],y_2, \dots, y_n\bigr] = \sum^n_{i=1} \bigl[x_1, \dots, x_{i-1}, [x_i,y_2, \dots, y_n], x_{i+1}, \dots, x_n\bigr]. \] A normalizer of a subspace \(X \subseteq A\) is the set \(N(X)\) of all elements \(a\in A\) such that \([a,x_2, \dots, x_n]\) \(\in X\) for all \(x_2, \dots, x_n\in X\). A nilpotent subalgebra \(H\) in \(A\) is a Cartan subalgebra if \(H=N(H)\). Assume that \(k\) is an algebraically closed field of characteristic zero and \(A\) has finite dimension over \(k\). Then any two Cartan subalgebras in \(A\) are conjugate by an automorphism of the form \(\exp(R_{x_2, \dots, x_n})\), where \(R_{x_2, \dots, x_n}\) is a nilpotent operator \(a\mapsto [a,x_2, \dots, x_n]\).