Constructing 3-Lie algebras

3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional simple 3-Lie algebra $(A, [,,]_{ω, δ_0})$ and a non-simple 3-Lie algebra $(A, [,,]_{ω_1, δ})$ are constructed by Laurent polynomials $A=F[t, t^{-1}]$ and its involutions $ω$ and $ω_1$ and derivations $δ$ and $δ_0$. At last of the paper, we summarize the methods of constructing $n$-Lie algebras for $n\geq 3$ and provide a problem.

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