  It is well known that the compatible linear and quadratic Poisson brackets of
the full symmetric and of the standard open Toda lattices are restrictions of
linear and quadratic $r$-matrix Poisson brackets on the associative algebra
${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on
${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the
splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper
triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a
quadratic Poisson structure on the cotangent bundle
$T^*{\mathrm{GL}}(n,{\mathbb{R}})$. This complements the interpretation of the
linear $r$-matrix bracket as a reduction of the canonical Poisson bracket of
the cotangent bundle.
