  We develop generating technique for solutions of $U(1)^3 5D$ supergravity via
dimensional reduction to three dimensions. This theory, which recently
attracted attention in connection with black rings, can be viewed as consistent
truncation of the $T^6$ compactification of the eleven-dimensional
supergravity. Its further reduction to three dimensions accompanied by
dualisation of the vector fields leads to 3D gravity coupled sigma model on the
homogeneous space $SO(4,4)/SO(4)\times SO(4)$ or $SO(4,4)/SO(2,2)\times
SO(2,2)$ depending on the signature of the three-space. We construct a $8\times
8$ matrix representation of these cosets in terms of lower-dimensional blocks.
Using it we express solution generating transformations in terms of the
potentials and identify those preserving asymptotic conditions relevant to
black holes and black rings. As an application, we derive the doubly rotating
black hole solution with three independent charges. A suitable contraction of
the above cosets is used to construct a new representation of the coset
$G_{2(2)}/(SL(2,R)\times SL(2,R))$ relevant for minimal five-dimensional
supergravity.
