I've been struggling with a paragraph appearing in this book:
http://www.math.sunysb.edu/~kirillov/liegroups/liegroups.pdf
( Example 3.10 bottom half of page 31 ).
It says that "by theorem 3.7, elements of the form $ exp(tJ_{x}), exp(tJ_{y}), exp(tJ_{z}) $ generate SO(3). Where $ J_{x}, J_{y}, J_{z} $ are the usual basis vectors for the Lie Algebra so(3)
What I am struggling with is this:
First of all, I know that these elements do generate SO(3), this is sometimes called the Tait-Bryan angle parametrization of SO(3). What I don't understand, is how elements of the above form generate a neighbourhood of the identity, and how it follows from anything that the book has said.
I understand that he is trying to say, if they do generate a neighbourhood of the identity, then they will in turn generate the entire group, as SO(3) is connected. However, from the results of the book up to this section, I can only justify as much as saying every element in SO(3) is of the form $ exp(tJ) $ for some J in the Lie algebra (not necessarily one of the basis elements) This is because the exponential map for a compact and connected Lie group is surjective. This is result is also known as Euler's axis-angle parametrization of SO(3).
So, how is the book supposed to justify the statement:
"by Theorem 3.7, elements of the form $ exp(tJ_{x}), exp(tJ_{y}), exp(tJ_{z}) $ generate a neighbourhood of the identity"?
EDIT: Sorry, Theorem 3.7 is on the page before ( page 30 ). It gives properties of the exponential map ( it is a local diffeomorphism, commutes with lie group homomorphisms..et c )
I even tried to work out the BCH formula for the product of two exponentials, but still, it is not been clear to me.
Thank you
