How these Big Five "properties" where derived is that people thought there was some pattern to questionnaires about behaviour and they ran a PCA on it. They noticed that PCA tended to bring up five principal components, so they interpreted them as five dominant personality traits that are shared by all humans.
This is not very good statistics: they basically used PCA to identify latent variables. But PCA cannot find latent variables. There is some criticism of this, but very rarely because "The Big Five" is basically accepted truth by the psychological community nowadays. In 20 years from now they'll probably debunk it and then hold up some other "model" of "human personality" and then the cycle will repeat.
Here's one critical paper:
https://pubmed.ncbi.nlm.nih.gov/19946599/
Whence I quote:
Consider, for instance, the personality literature, where people have discovered that executing a PCA of large numbers of personality subtest scores, and selecting components by the usual selection criteria, often returns five principal components. What is the interpretation of these components? They are “biologically based psychological tendencies,”; and as such are endowed with causal forces (McCrae et al., 2000, p. 173). This interpretation cannot be justified solely on the basis of a PCA, if only because PCA is a formative model and not a reflective one (Bollen & Lennox, 1991; Borsboom, Mellenbergh, & Van Heerden, 2003).