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We review the two different approaches for symplecticity of linear multi-step methods (LMSM) by Eirola and Sanz-Serna, Ge and Feng, and by Feng and Tang, Hairer and Leone, respectively, and give a numerical example between these two approaches. We prove that in the conjugate relation G3λτ∘G1τ=G2τ∘G3λτ with G1τ and G3τ being LMSMs, if G2τ is symplectic, then the B-series error expansions of G1τ, G2τ...