The Ringel map embeds the extended positive Borel and is full Hall exactly in finite Dynkin type

If the vertex set is empty, the representation category has only its zero object, both presented halves are $K$, the torus is trivial, and the assignment is the identity; every assertion follows directly. Assume henceforth that the vertex set is nonempty. Let $S_{\nu }$ denote the numerical Serre-only positive half in [F6]. By [F3], its generator map to the actual integral-half specialization at the inverse parameter is an isomorphism; the symmetric Serre coefficients are unchanged by $\nu \leftrightarrow {\nu }^{-1}$. Compose with [F4] to obtain the Hall assignment $\varphi :S_{\nu }\to C_K$. It is surjective and isometric for a nondegenerate source form. If $\varphi (x)=0$, then for every $y$ in the same homogeneous component its source pairing is $(\varphi (x),\varphi (y){)}_G=0$, forcing $x=0$ by [F3]. Separate the finitely many homogeneous components for an arbitrary vector. Thus the positive Hall assignment is injective, with image exactly the composition algebra.

The source Borel has the actual normal-form vector space $K[{\mathbb{Z}}^{Q_0}]{\otimes }_K{S}_{\nu }$. Indeed, homogeneity of the Serre relations makes $x\mapsto {\nu }^{(\alpha ,\mathrm{deg}x)}x$ an automorphism of $S_{\nu }$ for every $\alpha$; on the displayed vector space define $(K_{\alpha }x)(K_{\beta }y)={\nu }^{-(\beta ,\mathrm{deg}x)}{K}_{\alpha +\beta }xy$ for homogeneous $x,y$. Bilinearity of the symmetric form makes this product associative: on a triple the exponents are $-(\beta ,\mathrm{deg}x)-(\gamma ,\mathrm{deg}x+\mathrm{deg}y)$ and $-(\gamma ,\mathrm{deg}y)-(\beta +\gamma ,\mathrm{deg}x)$, which agree. It satisfies all relations in [F6]. Conversely moving torus letters left in that presentation defines the inverse map; both composites fix every $E_i,K_i^{\pm 1}$. Thus this is an isomorphism of algebras, proving the source normal form directly.

By [F1] and [F6], the target has the matching vector space $K[{\mathbb{Z}}^{Q_0}]{\otimes }_{K}H$, in the same left-torus order. Under these normal forms the map of [F2] is the identity on the torus factor tensor $\varphi$ of step 1.1. Write an element as a finite sum ${\sum }_{\alpha }{K}_{\alpha }{x}_{\alpha }$. If its image is zero, independence of the distinct target toral basis factors gives $\varphi (x_{\alpha })=0$ for every $\alpha$, and step 1.1 gives $x_{\alpha }=0$. The source map is therefore injective. Its image and bialgebra compatibility are those in [F2]. Every object in the selected representation category has finitely many subobjects, since its underlying vector spaces are finite sets; hence its coproduct has the ordinary finite-support tensor target. This argument requires no nondegeneracy of a torus pairing.

The full-Hall criterion for the positive half is [F5]. The two normal forms show that adjoining the full torus preserves equality or properness of the positive subspaces. Hence the embedding is onto the full extended Hall algebra exactly in the same componentwise finite ADE cases, including disconnected and empty quivers. The nilpotent convention is retained for every oriented-cycle case.