I am working on Lie groups, and I have several difficulties to show that the identity component of SO(2,1) is the product of an euclidian rotation fixing a vector X and an hyperbolic rotation in a plane containing X.
In the proof I have, they say that if A belongs to the identity component of SO(2,1), X and AX are linearly independant, which I agree.
My problem is, they say AX and X span a plan in which the dot product (of signature (2,1) ) is hyperbolic, in the sense that X is a negative vector and (AX + (AX,X) ) is a positive vector.
But how can AX + (AX,X) be a vector? Isn't it a addition between a vector and a scalar, hence undefined? And if the proof I have contains a mistake, can someone give me another?
Thanks in advance!