Relation between Clebsch-Gordan Coefficients and Wigner $D$-Matrix
It is known that Clebsch-Gordan coefficients are those of a linear transformation from the product basis ${|j_1,j_2;m_1,m_2\rangle}{m_1\in {-j_1,...,j_1},m_2\in {-j_2,...,j_2}}$ to the coupled basis ${|j_1,j_2;J,M\rangle\rangle}{J\in {|j_1-j_2|,...,j_1+j_2},M\in {-J,...,J}}$ (the double ket to distinguish between the two basis). These two basis are orthonormal basis in the space $\mathscr{E}(j_1)\otimes\mathscr{E}(j_2)$ and that means that there is a unitatry operator $\hat U$ that transforms one basis to the other. The matrix elements of this operator are by construction the CG-Coefficients.
On the other hand a physical rotation $R$ is implemented on a ket in $\mathscr{E}(j_1)\otimes\mathscr{E}(j_2)$ by a unitary operator $\mathscr D (R)$.
Question:
Is there a physical rotation $R$ such that $\mathscr D (R) = \hat U$, where $\hat U$ is the change-of-basis operator $\hat U: {|j_1,j_2;m_1,m_2\rangle} \mapsto {|j_1,j_2;J,M\rangle\rangle}$ ?
My answer is no and here are my two arguments
1. 956
Diagonalisation of observables (namely $\mathbf J^2$ & $J_z$) is not possible by physical rotations
2. The map $\mathscr D :SO(3) \to SU((2j_1+1)(2j_2+1))$ may not be surjective, i.e. there are unitary operators $\hat{ \mathcal O} \in SU((2j_1+1)(2j_2+1))$ which do not correspond to physical rotations $R \in SO(3)$
These two arguments do not exclude each other but I hope that someone can make this mathematically more precise, I hope this question makes sense.
Added:
in other words are there e.g. Euler angles $\alpha, \beta,\gamma$ such that matrix elements are something like this $$ \mathscr D^{(J)}_{M',M}(\alpha, \beta,\gamma) = \langle j_1,j_2;m_1,m_2|j_1,j_2;J,M\rangle $$ I know the indices are not consistent, they do not make sense! but this may not be a reason to beleive that there is no $R(\alpha, \beta,\gamma)$ whose representation $\mathscr D(R)$ does as
$$\hat U: {|j_1,j_2;m_1,m_2\rangle} \mapsto {|j_1,j_2;J,M\rangle\rangle}$$ 956