Garrett my guess is:
[tex]
L_{\vec{\xi_A}} \vec{\xi'_B} = C_{ABC} \vec{\xi'_C} = 2 \epsilon_{ABC} \vec{\xi'_C}
[/tex]
and
[tex]
L_{\vec{\xi'_A}} \vec{\xi_B} = C_{ABC} \vec{\xi_C} = 2 \epsilon_{ABC} \vec{\xi_C}
[/tex]
Now let's take even more risks (of doing false statements) postulating that...
[tex]
L_{\vec{\xi_A}} \vec{\xi'_B} = [{\vec{\xi_A}}, \vec{\xi'_B}]
[/tex]
Could we say then that the adjoint representation and the lie bracket are actually the same thing (homomorphic) ?!
[tex]
L_{\vec{\xi_A}} \vec{\xi'_B} = [{\vec{\xi_A}}, \vec{\xi'_B}]=ad({\vec{\xi_A})(\vec{\xi'_B})={ad_{\vec{\xi_A}}{\vec{\xi'_B}}
[/tex]
[itex]ad_{\vec{\xi_A}}[/itex] could be interpreted as a linear transformation of the vector field [itex]\vec{\xi_A}[/itex] that preserves a Lie bracket, [itex][{\vec{\xi_A}}, \vec{\xi'_B}][/itex] in this case.
Question 1: Is it true that the adjoint representation of su(2) is so(3)... and that the adjoint representation of su(2) give the structure constants which are also the matrix element of so(3). How so ?!
Question 2: What is the signification of 2 and 2 in [itex]2 \epsilon_{ABC} \vec{\xi'_C}[/itex] and [itex]2 \epsilon_{ABC} \vec{\xi_C}[/itex] ?
They are probably structure constants coefficients but are they matrix element... of which matrix ?
