Classification of 3-graded causal subalgebras of real simple Lie algebras

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 $\tau(h) = h$,weclassifytheLiealgebras$\mathfrak{g}(W,\tau,h)$ 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.