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Two types of higher order Lie $\ell$-ple systems are introduced in thispaper. They are defined by brackets with $\ell > 3$ arguments satisfyingcertain conditions, and generalize the well known Lie triple systems. One ofthe 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 specifictypes of $k$-Leibniz algebras, relevant in the construction, are introduced aswell. Both higher order Lie $\ell$-ple generalizations reduce to the standardLie triple systems for $\ell=3$.

k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems