Possible derivation of some $SO(p,q)$ group representations by means of a canonical realization of the $SO(p,q)$ Lie algebra

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1. [1] A general formula is given in H. Bacry, Space time and degrees of freedom of the elementary particle, Comm. Math. Phys., t. 5, 1967, p. 97, for the semi-simple Lie groups. Let us recall it: 1/2(d - r) = g where d, r, g denote respectively the dimensionality of the Lie group, the number of its fundamental invariants, and the number of the generators of a maximal abelian subalgebra of the enveloping algebra. This formula comes from a more general framework in Gelfand and Kirillov, Sur les corps liés avec algèbres enveloppantes des algèbres de Lie, Publications Mathématiques, n° 31 (I. H. E. S.). 
2. [2] [n/2] denotes the rank of the group, namely n/2 if n is even, (n — 1)/2 if n is odd. 
3. [3] Let us note the approach of A. KIHLBERG in which he considers as subgroup the maximal compact group and enters in similar considerations ( A. Kihlberg, Arkiv für Fysik, t. 30, 1965, p. 121). Many references on the study of particular non compact rotation groups are given in this paper. We recall some particular works in ref. [4]. Zbl0171.11703
4. [4] The DE SITTER group SO(4,1) has been treated thoroughly by Dixmier, in J. Bull. Soc. Math. France, t. 89, 1961, p. 9. The SO(3,2) group has been investigated by J.B. Ehrman, in Proc. Camb. Phil. Soc., t. 53, 1957, p. 290. Zbl0078.29302MR140614
5. [5] E.P. Wigner, Ann. Math., t. 40, 1939, p. 149. See also [6]. Zbl0020.29601JFM65.1129.01
6. [6] F.R. Halpern and E. Branscomb, Wigner's analysis of the unitary representations of the Poincaré group, UCRL-12359, 1965. 
7. [7] Y. Murai, Progr. Theor. Phys., t. 11, 1954, p. 441. H. Bacry, Ann. Inst. H. Poincaré, A II, 1965, p. 327. 
8. [8] In the following, the latin indices take always these values. 
9. [9] R. Raczka, N. Limic and J. Niederle, Discrete degenerate representations of non compact rotation group, IC/66/2, Trieste and Continuous degenerate representations of non compact rotation groups, IC/66/18, Trieste. Zbl0158.45805
10. [10] L. Castell, The physical aspects of the conformal group SO0(4,2), IC/67/66, Trieste. Zbl0162.58604
11. [11] H. Bacry and J.L. Richard, Partial theoretical group treatment of the relativistic hydrogen atom, 1967 (to be published in J. Math. Phys.). Zbl0158.45905
12. [12] I.M. Gel'fand, R.A. Minlos et Z.Y. Shapiro, Representations of the rotation and Lorentz groups and their applications, Pergamon Press, 1963. For other references, see [6]. Zbl0108.22005