The main result of this paper is to prove that if a (right) Leibniz algebra L is right nilpotent of degree

n, then L is strongly nilpotent of degree less or equal to 4n2 − 2n + 1. Résumé Nous prouvons

que toute algèbre de Leibniz (droite) L nilpotente à droite d'indice n est fortement nilpotente

d'un indice inférieur ou égal à 4n2 − 2n + 1 … Keywords. Leibniz algebra, right nilpotency, left

nilpotency, nilpotency, strong nilpotency, index. 2010 Mathematics Subject Classification:

17A32, 17B30 … In [1] it is proved that a Malcev algebra is strongly nilpotent if and only if it

is right nilpotent. So for Malcev algebras right nilpotency, left nilpotency and strong nilpotency

are equivalent to nilpotency. Since Malcev algebra is anti-commutative, right nilpotency and

left nilpotency are equivalent. This result fails for Leibniz algebras, see for example [4, Exemple

3.3], which is left nilpotent and not right nilpotent … ∗: bere_jean0@yahoo.fr † …