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New coset matrix for D=6 self-dual supergravity

Toroidal reduction of minimal six-dimensional supergravity, minimal five-dimensional supergravity and four-dimensional Einstein-Maxwell gravity to three dimensions gives rise to a sequence of cosets $O(4,3)/(O(4)\times O(3))\supset G_{2(2)}/(SU(2)\times SU(2))\supset SU(2,1)/S(U(2)\times U(1))$ which are invariant subspaces of each other. The known matrix representations of these cosets, however, are not suitable to realize these embeddings which could be useful for solution generation. We construct a new representation of the largest coset in terms of $7\times 7$ real symmetric matrices and show how to select invariant subspaces corresponding to lower cosets by algebraic constraints. The new matrix representative may be also directly applied to minimal five-dimensional supergravity. Due to full O(4,3) covariance it is simpler than the one derived by us previously for the coset $G_{2(2)}/(SU(2)\times SU(2))$.