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In this paper, we consider a Lie group $G$ equipped with two left-invariantRiemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannianmetrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangentLie group $TG$. The Levi-Civita connection, tensor curvature, and sectionalcurvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, wegive a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally,motivated by the recent work of D. N. Pham, using symplectic forms $\omega _1$and $\omega _2$ on $G$ we define a symplectic form $\tilde{\omega}$ on $TG$.

Riemannian geometry of tangent Lie groups using two left invariant Riemannian metrics