In terms of differential geometry, so(3) is the tangent space at the identity element of SO(3), with a Lie bracket [itex](X,Y)\mapsto[X,Y][/itex] defined by setting [X,Y] equal to the commutator at the identity, of the left-invariant (or right-invariant) vector fields corresponding to X and Y. (Those are the fields defined in the quote below. The two Lie algebras defined this way are isomorphic, so it doesn't matter if we use left multiplication or right multiplication to define the Lie bracket).
In terms of matrices, SO(3) is the set of 3×3 orthogonal matrices with determinant 1, and so(3) is the set of matrices X such that exp(tX) is in SO(3) for all real numbers t. (This is a trick to find a Lie algebra that's isomorphic to the group's Lie algebra. It works for matrix Lie groups). So it's the set of all antisymmetric traceless 3×3 matrices. (Mathematicians use the term "skew symmetric" or just "skew" instead of "antisymmetric". It means X
T=-X. "Traceless" means Tr X=0, where Tr X is the sum of the diagonal elements).
 Quote by Fredrik
Suppose that M and N are manifolds and [itex]\phi:M\rightarrow N[/itex] is a diffeomorphism. Then we can define a function
[tex]\phi(p)_*:T_pM\rightarrow T_{\phi(p)}N[/tex]
for each [itex]p\in M[/itex].
A Lie group is both a group and a manifold. We can use any member g of a group G to construct two diffeomorphisms [itex]\rho_g[/itex] and [itex]\lambda_g[/itex] that map G onto itself:
[tex]\rho_g(h)=hg[/tex]
[tex]\lambda_g(h)=gh[/tex]
The Lie algebra associated with the Lie Group is defined as the tangent space at the identity element, with a Lie bracket that will be defined below. Let's use the notation [itex]\mathfrak{g}=T_eG[/itex]
We can use either right or left multiplication to map the Lie algebra onto the tangent space at any other point g:
[tex]\rho_g_*:\mathfrak{g}\rightarrow T_gG[/tex]
[tex]\lambda_g_*:\mathfrak{g}\rightarrow T_gG[/tex]
Let's simplify the notation a bit:
[tex]\rho_g_*(L)=Lg[/tex]
[tex]\lambda_g_*(L)=gL[/tex]
We can use these maps to construct two vector fields [itex]X_L^\rho[/itex] and [itex]X_L^\lambda[/itex] for each vector L in the Lie algebra:
[tex]X_L^\rho|_g=Lg[/tex]
[tex]X_L^\lambda|_g=gL[/tex]
Either of these two vector fields can be used to define a Lie bracket on the Lie Algebra:
[tex][K,L]=[X_K^\rho,X_L^\rho]_e[/tex]
[tex][K,L]=[X_K^\lambda,X_L^\lambda]_e[/tex]
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