Relations for the difference of two dilogarithms

In this work, we propose a double-series representation of the difference between two dilogarithms with specific arguments. The summation is derived combining a formula we recently found for the so-called Grotendieck-Krivine constant, and an identity obtained by Lima for the difference of two dilogarithms with arguments $\sqrt{2}-1$ and $1-\sqrt{2}$ respectively. We also give an integral representation of the difference considered by Lima, on the basis of a formula published by Brychkov.