## The real Jacobi group G^J_1 revisited

The real Jacobi group G^{J}_{1}(R), defined as the semidirect product of group SL with the Heisenberg group H_{1}, is embedded in the 4x4 matrix realisation of the group Sp(2,R). The left-invariant one-forms on G^{J}_{1}(R) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x,y,θ,p,q,κ), and the left-invariant metric is obtained. The Heisenberg group H_{1} and Sp(2,R) are considered as subgroups of SL. An invariant metric in the variables (x,y,θ) on the Sasaki manifold SL is presented. The well known Kaehler balanced metric in the variables (x,y,p,q) of the 4-dimensional Siegel-Jacobi upper half-plane X^{J}_{1}=G^{J}_{1}(R)/[SO(2)×R]≈X_{1}×R^{2} is written down as sum of the squares of 4 invariant one-forms, where the Siegel upper half-plane X_{1} is realized as C∋τ:= x+iy, y>0. A 5-dimensional manifold, called extended Siegel-Jacobi upper half-plane X^{J}_{1}=G^{J}_{1}(R)/SO(2)≈X_{1}×R^{3},is considered, and its left-invariant metric in the variables (x,y,p,q,κ) is determined. As a bi-product, we directly prove that the Siegel-Jacobi upper half-plane X^{J}_{1} is not a naturally reductive space with respect to the balanced metric, but we underline that it is naturally reductive in the coordinates furnished by the FC-transform.