Double derivations of n-Lie algebras

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎with certain constraints on the base field then the centralizer of $ad(L)$ in $mathcal D(L)$ is‎ ‎trivial and $mathcal D(L)$ is centerless‎. ‎In addition‎, ‎we obtain that for every perfect $n$-Lie‎ ‎algebra $L$ with zero center‎, ‎the triple derivations of the derivation algebra $mathcal Der(L)$ are exactly‎ ‎the derivations of $mathcal Der(L)$‎, ‎and the triple derivations of the inner derivation algebra $ad(L)$ are‎ ‎precisely the derivations of $ad(L)$‎.