Undergrad Lie groups left invariant vector fields

[Mikeey aleex]

hello every one .
can someone please find the left invariant vector fields or the generator of SO(2) using Dr. Frederic P. Schuller method ( push-forward,composition of maps and other stuff)
Dr Frederic found the left invariant vector fields of SL(2,C) and then translated them to the identity using certain chart which covers the identity .
the lecture .

 

[Mikeey aleex]

The details are all in that video , the problem is that i can't find a single parameterization such that there exists some slots in the matrix ( e.g. SO(3) ) such that the parameters lying in each slot are independent of each other , the euler angles parameterization of SO(3) gives rise that each slot has a parameter as a function or has more the one parameter ( there is no independency between parameters in that matrix so i can't find a single coordinate chart in R^3 that covers the identity so i would do my calculations at the identity ), that why i can't use the dr. Frederic P. Schuller's method of finding the lie algebra of SO(3) at the identity using the push-foward of left invariant vector field

 

[RockyMarciano]

The details are all in that video , the problem is that i can't find a single parameterization such that there exists some slots in the matrix ( e.g. SO(3) ) such that the parameters lying in each slot are independent of each other , the euler angles parameterization of SO(3) gives rise that each slot has a parameter as a function or has more the one parameter ( there is no independency between parameters in that matrix so i can't find a single coordinate chart in R^3 that covers the identity so i would do my calculations at the identity ), that why i can't use the dr. Frederic P. Schuller's method of finding the lie algebra of SO(3) at the identity using the push-foward of left invariant vector field

In your first post you mentioned SO(2), now you switch to SO(3), a very different case, and you want to use an example from SL(2,C) that is complex and not even compact so even harder to compare to the previous cases, could you decide which group you want to work with?
In the meantime from the information in your last post I can tell you that the Lie algebra of SO(2) is ℝ, while that of SO(3) is so(3), not $ℝ^3$, SO(3) is not simply connected, so there you have a first problem for finding the parametrization you want in $ℝ^3$. You can find more details about what seems to be your concern here.

 

[Mikeey aleex]

my friend Rocky Marciano , I'm working on this method for SO(2) , SU(2) ,SO(3) AND SO(3,1) . i found all the lie algebras of each manifold by approximation method , states that there exists a small parameter (e)[(infinitesimal change)] in R^1 close to the Identity of the gruop (e.g. ) G Such that an element of G close to the identity is defined as g = I(identity of the group ) + eH
such that H represents the tangent vector space of G at the identity (i.e. lie algebra of G) , ( using (e.g. euler angles for SO(3) ) ) we can differentiate the parameterized element of the group (SO(3)) [ with is closed to the identity ( infinitesimal change ) ] with respect to the parameter (e) then evaluating at (0) so we can find H ( Lie algebra of SO(3) )