The Complete List of Solutions to M\"{o}bius's Exponential Equation

M"{o}bius addition is a non-associative binary operation defined on the complex open unit disk $\mathbb{D} = \{z\in\mathbb{C}:|z|<1\}$ by $a\oplus_M b = \dfrac{a+b}{1+\overline{a}b}$, and M"{o}bius's Exponential Equation is a non-linear functional equation of the form $L(a\oplus_M b) = L(a)L(b)$, where $L$ is a complex-valued function defined on $\mathbb{D}$. In [Aequat. Math. \textbf{91} (2017), 491--503], the authors address the problem of determining the solutions to M"{o}bius's Exponential Equation. In this paper, we determine the complete list of solutions to M"{o}bius's Exponential Equation using an algebraic approach.