[math] \newcommand{\mathds}{\mathbb}[/math]

Work out the Tannakian duality for the closed subgroups

[[math]] G\subset O_N [[/math]]

first as a consequence of the general results that we have, regarding the closed subgroups

[[math]] G\subset U_N [[/math]]

and then independently, by pointing out the simplifications that appear in the real case.