Example of a cluster algebra of geometric type and ground field

In most definitions of a cluster algebra of geometric type, it is said to be the subalgebra of the field $\mathbb Q (x_1,\cdots,x_n)$ generated by the cluster variables.

I would like to compute such an algebra but I am not sure of the result. As an example, let's try with the cluster algebra of rank $2$ associated with the seed $((x,y),B)$ where $B$ is the skew-symetric matrix $\begin{pmatrix} 0 & 1 \\-1 & 0\end{pmatrix}$.

There is only a finite number of clusters : $x,y,\dfrac{1+y}{x},\dfrac{1+x}{y},\dfrac{1+x+y}{xy}$ but what is the associated cluster algebra ? What are elements of it ?

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As you already stated, the clusteralgebra is always the $\mathbb{Q}$-subalgebra of the field $\mathbb{Q}(x_1,\dots,x_n)$ generated by the cluster variables.
So, in your case it is the subalgebra of $\mathbb{Q}(x,y)$ generated by $x,y,\dfrac{1+y}{x}, \dfrac{1+x}{y},\dfrac{1+x+y}{xy}$. This is sometimes denoted $\mathbb{Q}\left[x,y,\dfrac{1+y}{x}, \dfrac{1+x}{y},\dfrac{1+x+y}{xy}\right]$. So, for example this includes $1,x,y \dfrac{1+y}{x}, \dfrac{(1+y)^2}{x^2}, \dfrac{1+y}{x}+\dfrac{1+x}{y}$. But for example, $\frac{1}{1+x}$ is not an element of this algebra.