  Let $(\mathfrak{g},\tau)$ be a real simple symmetric Lie algebra and let $W
\subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and
generating with $\tau(W) = -W$. For elements $h \in \mathfrak{g}$ with $\tau(h)
= h$, we classify the Lie algebras $\mathfrak{g}(W,\tau,h)$ which are generated
by the closed convex cones \[C_{\pm}(W,\tau,h) := (\pm W) \cap
\mathfrak{g}_{\pm 1}^{-\tau}(h),\] where $\mathfrak{g}^{-\tau}_{\pm 1}(h) :=
\{x \in \mathfrak{g} : \tau(x) = -x, [h,x] = \pm x\}$. These cones occur
naturally as the skew-symmetric parts of the Lie wedges of endomorphism
semigroups of certain standard subspaces. We prove in particular that, if
$\mathfrak{g}(W,\tau,h)$ is non-trivial, then it is either a hermitian simple
Lie algebra of tube type or a direct sum of two Lie algebras of this type.
Moreover, we give for each hermitian simple Lie algebra and each equivalence
class of involutive automorphisms $\tau$ of $\mathfrak{g}$ with $\tau(W) = -W$
a list of possible subalgebras $\mathfrak{g}(W,\tau,h)$ up to isomorphy.
