  We prove that the universal covering group
$\widetilde{\mathrm{Sp}}(2,\mathbb{R})$ of $\mathrm{Sp}(2,\mathbb{R})$ does not
have the Approximation Property (AP). Together with the fact that
$\mathrm{SL}(3,\mathbb{R})$ does not have the AP, which was proved by Lafforgue
and de la Salle, and the fact that $\mathrm{Sp}(2,\mathbb{R})$ does not have
the AP, which was proved by the authors of this article, this finishes the
description of the AP for connected simple Lie groups. Indeed, it follows that
a connected simple Lie group has the AP if and only if its real rank is zero or
one. By an adaptation of the methods we use to study the AP, we obtain results
on approximation properties for noncommutative $L^p$-spaces associated with
lattices in $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$. Combining this with
earlier results of Lafforgue and de la Salle and results of the second named
author of this article, this gives rise to results on approximation properties
of noncommutative $L^p$-spaces associated with lattices in any connected simple
Lie group.
