  Two types of higher order Lie $\ell$-ple systems are introduced in this
paper. They are defined by brackets with $\ell > 3$ arguments satisfying
certain conditions, and generalize the well known Lie triple systems. One of
the generalizations uses a construction that allows us to associate a
$(2n-3)$-Leibniz algebra $\fL$ with a metric $n$-Leibniz algebra $\tilde{\fL}$
by using a $2(n-1)$-linear Kasymov trace form for $\tilde{\fL}$. Some specific
types of $k$-Leibniz algebras, relevant in the construction, are introduced as
well. Both higher order Lie $\ell$-ple generalizations reduce to the standard
Lie triple systems for $\ell=3$.
