The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere ${sum}$(0, 3), a one-holed torus ${sum}$(1, 1), and a four-holed sphere ${sum}$(0, 4). For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product <,> which gives the information about locations of axes of loxodromic elements. The signs of traces of holonomy elements, which are calculated by lifting a representation from PSL(2, $mathbb{C}$) to SL(2, $mathbb{C}$), play a very important role in determining the discreteness of holonomy groups.