TITLE: Exceptional isomorphisms of Lie groups
QUESTION [16 upvotes]: It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment of such isomorphims. By "systematic", I mean a presentation of a list of the most relevant of them (in the real and complex case, for instance) with some proofs and explanations. So, my question is, Does there exists any relatively complete reference on this topic?
Thanks in advance

REPLY [12 votes]: The question goes back to paper A:  Cartan's 1914 classification of real simple Lie algebras (Collected papers, Part I, Vol.1) with essentially complete answers for the Lie algebras . A complete discussion together with explicit isomorphisms for all the cases (both the Lie algebras and the Lie groups ) is in my book
  B:  Diff. Geometry, Lie Groups and Symmetric Spaces,  pp. 517- 528.
Comments: On the first page of A (bottom) Cartan states that real forms of the same simple Lie algebra over C are in general commpletely determined by the signature of the Killing form. As shown in B  p. 517 this is contradicted by so*(18) and so(12,6) as well as by su*(14) and su(9,5). Cartan's phrase "general" clearly and deliberately meant "almost all the time".  However, at the end of the paper (p. 352-355)  he identifies real forms on the basis of the signature statement (which would require some additional caution). As mentioned in B, p. 520 the isomorphism so*(8) ~ so(6,2) does not occur in Cartan's original list although in a later paper he shows that the corresponding symmetric spaces are isometric.
Generally the group isomorphism in B  are computational although some have geometric interpretation like the isomorphism so(3)~su(2) which comes from the standard stereographic projection.
S. Helgason