A note on the Jordan decomposition (Q2894394)

The multiplicative Jordan decomposition presents a linear automorphism of \(\mathbb{R}^n\) as a product of its elliptic, hyperbolic and unipotent components. For a Lie group \(G\) one can define an abstract Jordan decomposition of an element \(g\in G\) by taking the Jordan decomposition of the adjoint map \(\mathrm{Ad}(g)\). In the paper under review the authors give an elementary proof of the fact that for real algebraic Lie groups the abstract Jordan decomposition and the multiplicative Jordan decomposition coincide. The main step towards this is to show that the elliptic and the hyperbolic components of both the additive and the multiplicative Jordan decompositions are given as polynomials of the original operator and that the same happens for the unipotent component. As a byproduct the authors get that a semi-simple linear Lie group is the connected component of the identity of an algebraic group, hence closed.