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In many places I have seen the SU(2) and SO(3) lie algebras used interchangeably. How are they exactly identical? Moreover, what about their lie groups? Are they identical as well. It would be great if you could give me both algebraic, and geometric reasons.

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en.wikipedia.org/wiki/… –  NiftyKitty95 Nov 28 '12 at 9:13
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Here is a fact concerning the lie groups you mentioned.

Namely $SU(2)$ is isomorphic (as a lie group) to $S^3$ and is a (universal) double cover of $SO(3)$. So since this covering map is a local diffeomorphism you can see that their lie algebras should coincide.

Note also that $SO(3)$ of course is diffeomorphic to $\mathbb{R}P^3$ and this double cover is just the usual one you know from projective spaces.

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