## On some causal and conformal groups.(English)Zbl 0873.17030

Let $$V$$ be a simple Euclidean Jordan algebra, $$V^c$$ its conformal compactification and Co$$(V)$$ the Kantor-Koecher-Tits group of conformal transformations of $$V$$. Consider an involutive transformation on the set of invertible elements of the Jordan algebra of the form $$x\mapsto (\alpha(x))^{-1}$$ and its induced involution on Co$$(V)$$, where $$\alpha$$ is in the structure group of $$V$$. Denote by $$G$$ the corresponding fixed point subgroup of Co$$(V)$$ and by $$X=G\cdot 0$$ its orbit in $$V^c$$. $$X$$ is then a symmetric space and is also called a Makarevich space. They were classified by B. O. Makarevich [Math. USSR, Sb. 20, 406-418 (1974); translation from Mat. Sb., Nov. Ser. 91(133), 390-401 (1973; Zbl 0279.53047)] without proofs. The author gives a proof by using the result of K. H. Helwig [Halbeinfache reelle Jodan-Algebren, Habilitationsschrift, München 1967] on the classification of involutive automorphisms of Jordan algebras. The author also identifies the pseudogroup of causal transformations of the space $$X$$ with the group $$\text{Co}(V)$$, some examples being worked out for the classical matrix Jordan algebra by elementary methods.