Oscillator ladder-operator sign convention ($a_{\pm }\mapsto -a_{\pm }$)

For real coefficients in $a_{\pm }=c_{\pm }X+i{d}_{\pm }P$, imposing $a_{-}^{†}=a_{+}$, $[a_{-},a_{+}]=I$ and $H=\hslash \omega (a_{+}{a}_{-}+\lambda I)$ givesIndeed the adjoint operator condition makes the two $c$ coefficients equal and the two $d$ coefficients opposite. The canonical commutation relation gives $2\hslash c_{-}{d}_{-}=1$, and matching the coefficients of $X^2$ and $P^2$ fixes their magnitudes. Both signs change together. The conventional creation and annihilation operators choose $\epsilon =1$; the other sign is an equivalent phase convention for the number states.