FullSimplify[Reduce[{-((6 [Alpha]^4 + 21 [Alpha]^3 [Lambda] + 27 [Alpha]^2 [Lambda]^2 + 15 [Alpha] [Lambda]^3 + 3 [Lambda]^4 + Sqrt[3] Sqrt[-([Alpha] + [Lambda])^6 (-72 + 16 [Alpha]^3 [Lambda] + 21 [Lambda]^2 + 4 [Alpha] [Lambda] (-9 + 4 [Lambda]^2) + 4 [Alpha]^2 (-15 + 8 [Lambda]^2))])/( 4 ([Alpha] + [Lambda])^4)) < 0 && -((6 [Alpha]^4 + 21 [Alpha]^3 [Lambda] + 27 [Alpha]^2 [Lambda]^2 + 15 [Alpha] [Lambda]^3 +  3 [Lambda]^4 - Sqrt[3] Sqrt[-([Alpha] + [Lambda])^6 (-72 + 16 [Alpha]^3 [Lambda] + 21 [Lambda]^2 + 4 [Alpha] [Lambda] (-9 + 4 [Lambda]^2) +4 [Alpha]^2 (-15 + 8 [Lambda]^2))])/( 4 ([Alpha] + [Lambda])^4)) < 0 && -((3 (-2 [Beta] + [Lambda] + [Lambda] [Mu]))/([Alpha]
+ [Lambda])) < 0 && [Alpha] ([Alpha] + [Lambda]) > -3 && [Lambda]
([Lambda] + [Alpha]) > 3}, {[Alpha], [Beta], [Lambda], [Mu]},
Reals, Cubics -> True, Quartics -> True]]