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While I understand that their corresponding lie algebras su(3) and so(2) have the same commutator relations

[tex] \mbox{SO(3)}: \left[ \tau^i, \tau^j\right] = \iota \varepsilon_{ijk} \tau^k[/tex]

[tex] \mbox{SU(2)}: \left[ \frac{\sigma^i}{2}, \frac{\sigma^j}{2}\right] = \iota\varepsilon_{ijk} \frac{\sigma^k}{2}[/tex]

so that the structure constants of each are identical, and as each Lie Algebra is uniquely defined by it structure constants, both algebras are identical.

The elements being operated on are, clearly, very different. If I were to distinguish the two groups in a discussion, I cannot speak of Algebra as the Algebra is purely the rule of composition between elements of the group(not the elements themselves)?. The Algebras are truly identical.

To distinguish, I would say

"Both Groups obey the same Lie Algebra, but whose infinitesimal generators are different"

yes?

Thanks

edit: I did search the forum, but could only find much more general questions about Lie Algebras so I decided to make a new topic.