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For example, the commutation relations:

[itex][J_x,J_y]=i\sqrt{2} J_z [/itex]

[itex][J_y,J_z]=\frac{i}{\sqrt{2}} J_x [/itex]

[itex][J_z,J_x]=i\sqrt{2} J_y [/itex]

is actually SO(3), as can be seen by redefining [itex]J'_x =\frac{1}{\sqrt{2}} J_x [/itex]: then [itex]J'_x [/itex], [itex]J_y [/itex] and [itex]J_z [/itex] have the SO(3) algebra. So the commutation relations above is SO(3). But how do we know that just by looking at it?

When you start taking linear combinations of generators, including sometimes with complex coefficients as in [itex] J_x+iJ_y[/itex], how can you tell the resulting commutators are SO(3)?