These are the earliest citations I could find for the phrase "loss of generality" in JSTOR. Note how they all slightly differ from the strict "without loss of generality" form. Also note how they're all from William R. Hamilton.
... and many of the new partial differential coefficients vanish, without producing, by this simplification, any real loss of generality
- Third Supplement to an Essay on the Theory of Systems of Rays, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 17, (1831), pp. v-x, 1-144
Mr. Jerrard has therefore accomplished a very remarkable simplification of this general problem, since he has reduced it to the problem of discovering two real functions of two arbitrary real quantities, by showing that, without any real loss of generality, it is permitted to suppose ...
- On the Argument of Abel, Respecting the Impossibility of Expressing a Root of Any General Equation above the Fourth Degree, by Any Finite Combination of Radicals and Rational Functions, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 18, (1839), pp. 171-259
And if we farther simplify the formulae by supposing $ a = 1, b=0, c=0, d=0$, which will be found in the applications to involve no essential loss of generality ...
- Researches Respecting Quaternions. First Series, William Rowan Hamilton, The Transactions of the Royal Irish Academy, Vol. 21, (1846), pp. 199-296