A direct derivation is presented for the Generalized Euler’s equation in the translated base system that is located at the center of mass of the body. The use of Lie algebra reveals an interesting property of the time-variant inertia matrix I(t) — its rate of change is the Lie product [Ω, I], where [Ω] is the skew-symmetric representation of the angular velocity ω. This observation also simplifies the derivation. The standard body system form and the ZRP-form are then deduced. With differing meanings of terms, all three forms have a similar appearance: MG = [I] α + [Ω] [I] ω.

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