Lagrange's theorem
is universal, because every positive integer can be written as a sum of 4 squares, by Lagrange's four-square theorem. By the 15 theorem, to verify this, it is sufficient to check that every positive integer up to 15 is a sum of 4 squares. (This does not give an alternative proof of Lagrange's theorem, because Lagrange's theorem is used in the proof of the 15 theorem.)