An infinite-dimensional 2-generated primitive axial algebra of Monster type

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α,β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\beta )$$\end{document}, over a field of characteristic other than 2, has dimension at most 8 if α∉{2β,4β}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \notin \{2\beta ,4\beta \}$$\end{document}. In this note, we show that Rehren’s bound does not hold in the case α=4β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =4\beta $$\end{document} by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,12)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,\frac{1}{2})$$\end{document} over an arbitrary field F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {F}}}$$\end{document} of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.