whose Coxeter-Dynkin diagram splits into two copies of $A_{2}$, and $E_{8}$, the largest exceptional Lie algebra, which has four $A_{2}$ building blocks as shown in Figure \[ADE8\_Dynkin\_Diagrams\]. As is well known, $A_{4}=SU(5)$ and $E_{8}$ are very important for unification physics. We will also discuss the exceptional Lie algebra $G_2$, which contains $SU(3)$ and has a fifth-root deformation with Fibonacci fusion rules.

Gauge Unification Physics Review {#gaugeunireview}
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Let us provide a short review of the Lie algebra unification physics program closely following the references [@Georgi; @Koca3; @Koca4]. The well known representation theory of $SU(2)$ can be extended to higher dimensional Lie groups and algebras. The structure of a simple Lie algebra is described by its root and weight systems. The classification of simple Lie algebras is an extensive subject on which we recommend the references [@Georgi; @Zee; @gilmore]. Here we collect and review just the elements necessary for our discussion. For example we have that the root system $A_n$ describes $SU(n+1)$, while $D_n$, with $n\geq3$, describes the Lie algebra of $SO(2n)$. The exceptional Lie algebras $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$ have their root systems with the same respective names. For our discussion we will focus on $A_n$, in particular 