Or who first solved the harmonic oscillator in the algebraic method?
(Almost certainly) Paul Dirac$^\dagger$.
Generations learned the algebraic derivation of the oscillator spectrum from Dirac's landmark book, The Principles of Quantum Mechanics. I am confident it is featured in his first edition (1930), but I only have access to his 3rd (1947), in which it is detailed in classic Diracese in Chapter IV, §34, p 136 et seq.
G Farmelo's 2009 biography (The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius) describes the invention having been made in the winter of 1926-27 in Copenhagen, and you may see it in p 251 (the "q-number" operators b) of his (1927)
The quantum theory of the emission and absorption of radiation, Proceedings of the Royal Society of London Ser A, 114(767), 243-265;
although he is steering the reader to his §8 of his (1926) The elimination of the nodes in quantum mechanics, Proceedings of the Royal Society of London Ser A 111 (757), 281-305.
It is evident he kept on streamlining and refining the concept/method for quite a while.
$\dagger$ Also in G Farmelo's celebrated biography of PAMD, p 118, it is stated that
At the end of January , as he was preparing to leave Copenhagen, Dirac posted his paper to the Royal Society. It turned out that he was the first to introduce the mathematics of creation and annihilation into quantum theory, though his results had been reached independently by John Slater, studying in Cambridge with Fowler. Slater was one of the many who admired Dirac’s paper for its content but found its presentation perversely complicated: ‘his paper was a typical example of what I very much distrusted, namely one in which a great deal of seemingly unnecessary mathematical formalism is introduced'.
This is meant to be referenced in Slater, J. (1975) Solid-State and Molecular Theory: A Scientific Biography, New York: John Wiley and Sons. The subtext is that Slater anticipated the radiation picture in parallel, however without the glorious ladder operator formalism asked about!