The $\mathrm{S}\mathrm{L}(2,\mathbb{C})$-character variety of $8_{18}$

The knot $8_{18}$ is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its $\mathrm{S}\mathrm{L}(2,\mathbb{C})$-character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of $\mathrm{S}\mathrm{L}(2,\mathbb{C})$.