At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\gamma$-matrices. I couldn't find anything more in the book cited there, either.
But what is the relation between these 6 sets? Are they related by unitary transformations or else? For example, we can think of the following to sets. $$\alpha_0=\sigma_z\otimes\tau_z,\alpha_1=\sigma_x\otimes\tau_0,\alpha_2=\sigma_y\otimes\tau_0,\alpha_3=\sigma_z\otimes\tau_x,\alpha_4=\sigma_z\otimes\tau_y$$ $$\beta_0=\sigma_z\otimes\tau_0,\beta_1=\sigma_x\otimes\tau_z,\beta_2=\sigma_y\otimes\tau_0,\beta_3=\sigma_x\otimes\tau_x,\beta_4=\sigma_x\otimes\tau_y$$