Construct an $SO(3)$ rotation inside two $SU(2)$ fundamental rotations We know that two $SU(2)$ fundamentals have multiplication decompositions, such that $$ 2 \otimes 2= 1 \oplus 3.\tag{1}$$
In particular, the 3 has a vector representation of $SO(3)$. While the 1 is the trivial representation of $SU(2)$.

I hope to see the precise $SO(3)$ rotation from the two $SU(2)$ fundamental rotations.


*

*


So let us first write two $SU(2)$ fundamental objects in terms of an $SO(3)$ object. In particular, we can consider the following three:

$$
|1,1\rangle= \begin{pmatrix}
1\\
0
\end{pmatrix}\begin{pmatrix}
1\\
0
\end{pmatrix}=  | \uparrow \uparrow \rangle,$$
$$|1,0\rangle
={1 \over \sqrt{2}  }
(\begin{pmatrix}
1\\
0
\end{pmatrix}
\begin{pmatrix}
0\\
1
\end{pmatrix}
+
\begin{pmatrix}
0\\
1
\end{pmatrix}
\begin{pmatrix}
1\\
0
\end{pmatrix})={1 \over \sqrt{2}  }(| \uparrow \downarrow \rangle+  |  \downarrow \uparrow \rangle)
,$$
$$|1,-1\rangle = \begin{pmatrix}
0\\
1
\end{pmatrix}\begin{pmatrix}
0\\
1
\end{pmatrix}=  |  \downarrow  \downarrow \rangle.
$$
where the $| \uparrow \rangle$ and $ \downarrow \rangle$ are in $SU(2)$ fundamentals. And we shothand
$| \uparrow \uparrow \rangle \equiv | \uparrow \rangle |\uparrow \rangle $ and so on.

question: How do we rotate between $|1,1\rangle$,  $|1,0\rangle$, $|1,-1\rangle$, via two $SU(2)$ rotations acting on two $SU(2)$ fundamentals? Namely, that is, construct an $SO(3)$ rotation inside the two $SU(2)$ fundamental rotations?
The $SU(2)$ has three generators, parametrized by $m_x,m_y,m_z$; how do we write down the generic $SO(3)$ rotations from two $SU(2)$ rotations?

Let us consider an example, an $SU(2)$ rotation $U$ acting on the $SU(2)$ fundamental $\begin{pmatrix}
1\\
0
\end{pmatrix}$ give rise to
$$
U \begin{pmatrix}
1\\
0
\end{pmatrix}=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2}) & (i m_x -m_y) \sin(\frac{\theta}{2}) \\
 (i m_x +m_y) \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2})-{i m_z} \sin(\frac{\theta}{2}) \\
\end{pmatrix}
\begin{pmatrix}
1\\
0
\end{pmatrix}=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
 (i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}
\equiv\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})
\begin{pmatrix}
1\\
0
\end{pmatrix}
+
 (i m_x +m_y) \sin(\frac{\theta}{2})
\begin{pmatrix}
0\\
1
\end{pmatrix}
$$
In other words, the $SU(2)$ rotation $U$ (with the $|\vec{m}|^2=1$) rotates $SU(2)$ fundamentals. Two $SU(2)$ rotations act as
$$
UU|1,1\rangle =
U   \begin{pmatrix}
1\\
0
\end{pmatrix}U   \begin{pmatrix}
1\\
0
\end{pmatrix}
=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
 (i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
 (i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}
$$

Hint: Naively, we like to construct
$$
L_{\pm} =L_{x} \pm i L_y,
$$ such that $L_{\pm}$ is an operator out of two $SU(2)$ rotations acting on two $SU(2)$ fundamentals, such that it raises/lowers between $|1,1\rangle$,  $|1,0\rangle$, $|1,-1\rangle$.


question 2: Is it plausible that two $SU(2)$ are impossible to perform such $SO(3)$ rotations, but we need two $U(2)$ rotations?

 A: The following solution originates from the theory of geometric quantization. I'll not explain the full theory behind it, but I'll give here the solution, then briefly discuss how to check that this is the required solution.
A general $SU(2)$ group element in the fundamental representation can be written as:
$$ g = \begin{bmatrix}
\alpha & \beta\\ 
 -\bar{\beta}& \bar{\alpha}
\end{bmatrix} $$
with
$$|\alpha|^2+\beta|^2=1$$
The three dimensional Hilbert space of the three dimensional representation can be parametrized by:
$$ \psi (z) = a + b z + cz^2 \quad (1) $$
where $x$ is an indeterminate
The action of $SU(2)$ on this vector space is given by:
$$ (g\cdot \psi)(z) = (-\bar{\beta} z + \bar{\alpha})^2 \psi( \frac{\alpha z + \beta}{-\bar{\beta} z + \bar{\alpha}}) \quad (2)$$


*

*To see that this is a representation, one can check that the composition of the action of two group elements coincides with the action of their product.

*To see that this is a faithful $SO(3)$ representation but not a faithful $SU(2)$, we can easily see that for the nontrivial element of the center:
$$ g_c = \begin{bmatrix}
-1 & 0\\ 
0 & -1
\end{bmatrix} $$
We have for every $\psi$
$$ (g_c\cdot \psi)(z) = \psi(z)$$

*Although, the "spherical" components $a, b, c$ are complex. To see that the representation is real, one can see that the "Cartesian" components $ (a+c), b, i^{-1}(a-c)$ transform by means of only real combinations of $\alpha$ and $\beta$.

A: Perhaps the following is helpful: 


*

*OP's eq. (1) is to be understood as a relation between complex representations of $SU(2)$, i.e complex vector spaces. Recalling that the fundamental $SU(2)$ representation ${\bf 2}\cong \overline{\bf 2}$ is isomorphic to the complex conjugate representation, let us instead consider the isomorphism
$$ {\bf 2}\otimes \overline{\bf 2}~\cong~{\bf 1}\oplus {\bf 3}. \tag{A}$$

*The left-hand side of eq. (A) can be realized as the real vector space $u(2)$ of $2\times 2$ Hermitian matrices. The group $SU(2)$ acts on $u(2)$ via conjugation. Given a spinor $| \psi\rangle\in {\bf 2}$, then $$ {\bf 1}\oplus su(2)~\cong~ u(2)~\ni~| \psi\rangle \langle\psi | ~=~\frac{1}{2}\sum_{\mu=0}^3x^{\mu} \sigma_{\mu}, \qquad (x^0,x^1,x^2,x^3) ~\in~\mathbb{R}^4. \tag{B}$$
The triplet ${\bf 3}$ corresponds to the traceless part, that is: $su(2)$. Hence the spinor $| \psi\rangle$ represents the 3-vector $\vec{r}=(x^1,x^2,x^3)$. See also this related Phys.SE post.
