Irreducible representation of so(3)

[nematic]

Hi guys, I have a question which is very fundamental to representation theory.
What I am wondering is that whether a first rank cartesian representation of so(3) is irreducible.
As I understand first rank cartesian representation of so(3) can be parametrized in terms of the Euler angles. That matrix representation of so(3) contains sine and cosine of the three Euler angles which are rotation matrices that transform a vector (x,y,z).
So my understanding is that representation must be irreducible because each matrix in so(3) transform a vector by rotating it around a direction. However each matrix in so(3) rotates a vector around a different vector. Therefore there is no invariant one and two dimensional subspace of so(3).
That seems to be a plausible explanation. What I think to be more solid proof is to be able to say that any similarity transformation in so(3) does not simultaneously sent all representative of so(3) into one block diagonal form.
I was wondering if it is already proved in the literature or is there any way we can prove it.

 

[fresh_42]

A one dimensional representation is per definition irreducible, since there cannot be subspaces. It is also a trivial representation since for all $X\in \mathfrak{g}$ where $\mathfrak{g}$ is a semisimple Lie algebra, and $\mathfrak{su}(3)$ is simple, ergo semisimple, we can find $Y,Z \in \mathfrak{g}$ such that $X=[Y,Z]$. Now we get $X.v=[Y,Z].v=Y.Z.v-Z.Y.v=\lambda(Y)\lambda(Z).v-\lambda(Z)\lambda(Y).v=0$ since every operation of $Y$ on $v$ turns $v$ into a multiple $\lambda(Y)\cdot v$ and $\lambda$ has to be a linear form.