I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of...
I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of Lie Algebras.
My problem is related to Lemma 6.1 (1st edition):
Let be the Killing form of a Lie algebra over , and the Killing form of the complexification Lie algebra . Then
Understanding the previous equation as I would get twice the result stated in the book. This is due to the following facts.
1. , acting as
therefore I can write it as a direct sum of the real adjoint map
1. Helgason's definition of complexification: with the complex structure
extending the Lie bracket by -linearity:
2. Killing form of any Lie algebra over an arbitrary field :
where is the -linear map .