Analytic continuation of representations between $SO(4)$ and $O(1,3)$ Lie algebras

Question

$SO(4)$ transformations and Lorentz transformations are isomorphic in a neighbourhood of their identity element. Can anyone shed light about how this could lead to the analytic continuation of their algebra representations?

Answer 1

Here, by way of explication of the WP discussion of the Lie algebras involed.

so(4) ~ so(3)×so(3) has these two ideals, whose generators are A and B, respectively. Taking (A+B)/2 and (A-B)/2, respectively, yields so(4) in the conventional real antisymmetric basis.

Taking, instead, J=(A+B)/2 and K = i (B-A)/2, yields so(1,3) of the Lorentz group, with its peculiar diag(-1,1,1,1) metric.

I have skipped anything relating to the global properties of the group, etc, since you only wanted the complexification part.