# Question about SO(3) and "infinitesimal generators" [duplicate]

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

• Could you tell us what theorem 3.7 is? Mar 10, 2013 at 19:53
• I made an edit. It's on the page directly before ( on page 30 ). I just lists the well known basic properties of the exp map. Thanks Mar 10, 2013 at 19:56