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We deﬁne the Lie algebra gβ = {α ∈ End(n) : [α, β] = 0, tr αβ = 0}, and take any reductive subgroup Gβ of GL(n) with Lie algebra gβ (existence is guaranteed by rationality of β). [·, ·] = g[g −1 ·, g −1 ·]. 1. [L. , Xn } of n consider β and Gβ ⊂ GL(n) as deﬁned above. [·, ·] is closed in Λ2 n∗ ⊗ n then n is an Einstein nilradical and β + ||β||2 I ∈ Der(n). [·, ·] is closed in Λ2 n∗ ⊗ n . 2. 1. Let n be the 4-dimensional 3-step nilpotent Lie algebra with Lie bracket given by [X1 , X2 ] = X3 + X4 , [X1 , X3 ] = X4 .

2) Rx, y = − 12 1 4 [x, xi ], xj [y, xi ], xj + ij [xi , xj ], x [xi , xj ], y , ij for all x, y ∈ s, where {xi } is any orthonormal basis of (s, ·, · ). It is proved in [L. 2) (see the notation in Section 3 and replace Rn by s). 4)). 1) for the Ricci operator is precisely the value of the moment map at the Lie bracket [·, ·] of s (up to scaling). 4) tr cI + 12 B + S(ad H) E = 1 4 π(E)[·, ·], [·, ·] . Let S be an Einstein solvmanifold with Ric = cI. We can assume that S is not unimodular by using [Dotti 82], thus H = 0 and tr ad H = ||H||2 > 0.

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