Commutator SO(3) - what am I doing wrong?

[BWV]

reading that the commutator of rotations on two orthogonal axes is i * the rotation matrix for the third axis

but if I commute this

\begin{pmatrix}\mathrm{cos}\left( \theta\right) & -\mathrm{sin}\left( \theta\right) & 0\cr \mathrm{sin}\left( \theta\right) & \mathrm{cos}\left( \theta\right) & 0\cr 0 & 0 & 1\end{pmatrix}

with this

\begin{pmatrix}1 & 0 & 0\cr 0 & \mathrm{cos}\left( \theta\right) & -\mathrm{sin}\left( \theta\right) \cr 0 & \mathrm{sin}\left( \theta\right) & \mathrm{cos}\left( \theta\right) \end{pmatrix}

I get (using innerproduct function in Maxima)
\begin{pmatrix}0 & \mathrm{cos}\left( \theta\right) \,\mathrm{sin}\left( \theta\right) -\mathrm{sin}\left( \theta\right) & -{\mathrm{sin}\left( \theta\right) }^{2}\cr \mathrm{cos}\left( \theta\right) \,\mathrm{sin}\left( \theta\right) -\mathrm{sin}\left( \theta\right) & 0 & \mathrm{cos}\left( \theta\right) \,\mathrm{sin}\left( \theta\right) -\mathrm{sin}\left( \theta\right) \cr {\mathrm{sin}\left( \theta\right) }^{2} & \mathrm{cos}\left( \theta\right) \,\mathrm{sin}\left( \theta\right) -\mathrm{sin}\left( \theta\right) & 0\end{pmatrix}

which has a determinant of zero and therefore not part of SO(3)

obviously I am not getting something, but don't see it. Any help is much appreciated

 

[qbert]

you are confusing the Lie group and the Lie algebra. The commutation
relation applies to the Lie algebra (ie the generators of the Lie group).

 

[BWV]

ah, thanks.