  Let $G_1$ be a semisimple real Lie group and $G_2$ another locally compact
second countable unimodular group. We prove that $G_1 \times G_2$ has fixed
price one if $G_1$ has higher rank, or if $G_1$ has rank one and $G_2$ is a
$p$-adic split reductive group of rank at least one. As an application we
resolve a question of Gaboriau showing $SL(2,\mathbb{Q})$ has fixed price one.
Inspired by the very recent work arXiv:2307.01194v1 [math.GT], we employ the
method developed by the author and Mikl\'os Ab\'ert to show that all
essentially free probability measure preserving actions of groups weakly factor
onto the Cox process driven by their amenable subgroups. We then show that if
an amenable subgroup can be found satisfying a double recurrence property then
the Cox process driven by it has cost one.
