Isomorphism between so(3) and su(2)

[MrRobot]

Homework Statement

How do I use the commutation relations of su(2) and so(3) to construct a Lie-algebra isomorphism between these two algebras?

Homework Equations

The commutation relations are [t^a, t^b] = i epsilon^abc t^c, the ts being the basis elements of the algebras. They basically have the same commutation relation, only t^a are two by two by the su(2) while 3X3 by so(3).

The Attempt at a Solution

 

[fresh_42]

How they are represented by matrices isn't important. If you have a linear, bijective mapping
$$φ : \mathfrak{su}(2) \longrightarrow \mathfrak{so}(3)$$
e.g. if you map all basis vectors ##t^α \longmapsto {s}^α## then you have to check whether ##φ([t^α,t^β]) = [φ(t^α),φ(t^β)] = [{s}^α,s^β].## If this is the case for all pairs ##(α,β)## then it is a Lie-algebra isomorphism.