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Let $E \rightarrow M$ be a smooth vector bundle with a bilinear product on$\Gamma(E)$ satisfying the Jacobi identity. Assuming only the existence of ananchor map $\mathfrak{a}$ we show that $\mathfrak{a}([X,Y]) =[\mathfrak{a}X,\mathfrak{a}Y]_c$. This gives the redundancy of the homomorphismcondition in the definition of Leibniz algebroid (in particular if it arisesfrom a Nambu-Poisson manifold); an aspect not addressed in the literature. Weapply our result to the brackets of Hagiwara, Ibanez et. al; we settle an oldquery of Uchino on redundancy for Courant bracket.

Bracket-Preserving property of Anchor Maps and Applications to Various Brackets