Why for the closed string we need to take both left and right moving modes into account when deriving the mass?

In the closed bosonic string we have both left and right moving sectors with modes $alpha_m^mu$ and $tilde{alpha}_m^mu$ respectively where $min mathbb{Z}$ and where we have $alpha_0^mu = tilde{alpha}_0^mu = frac{ell_s}{2}p^mu$. Now the Virasoro generators are given by

$$L_m=frac{1}{2}sum_{n=-infty}^infty alpha_{m-n}cdot alpha_n,quad tilde{L}_m=frac{1}{2}sum_{n=-infty}^infty tildealpha_{m-n}cdot tildealpha_ntag{1}$$

and classically the vanishing of the energy-momentum tensor requires $L_m = tilde{L}_m =0$. Now it is said in some references (for example Becker, Becker & Schwarz, page 40) that to get the mass formula $M^2$ for the classical closed string we need to take both left-moving and right-moving modes into account. I don’t get this, my impression is that we can obtain $M^2$ from just $L_0=0$ or $tilde{L}_0=0$. Indeed:

$$L_0 = frac{1}{2}sum_{n=-infty}^infty alpha_{-n}cdot alpha_n = frac{1}{2}alpha_0^2+frac{1}{2}sum_{n=-infty}^{-1}alpha_{-n}cdot alpha_n +frac{1}{2}sum_{n=1}^infty alpha_{-n}cdot alpha_n=-frac{ell_s^2}{8}M^2+sum_{n=1}^infty alpha_{-n}cdot alpha_ntag{2}$$

where in the third equality we have reindexed $m =-n$ in the first sum, used that the classical variables commute and used the definition of $alpha_0$. When we set $L_0 =0$ we are able to get $$M^2=frac{8}{ell_s^2}sum_{n=1}^{infty}alpha_{-n}cdot alpha_ntag{3}.$$

Now the same derivation in (2) with $tilde{L}_0$ gives $$M^2=frac{8}{ell_s^2}sum_{n=1}^infty tilde{alpha}_{-n}cdot tilde{alpha}_ntag{4},$$

and clearly if we sum them up and divide by two we find $$M^2=frac{4}{ell_s^2}sum_{n=1}^infty left(alpha_{-n}cdot alpha_n+tilde{alpha}_{-n}cdot tilde{alpha}_nright)tag{5}.$$

My impression is that we can express $M^2$ either in terms of just $alpha_m$ as in (3), equivalently in terms of just $tilde{alpha}_n$ as in (4) or in terms of both as in (5). But still, as I said, some references say that for the closed string we need both $L_0=0$ and $tilde{L}_0=0$ to derive the mass. What am I missing here? We do we need both $L_0=0$ and $tilde{L}_0=0$ to derive the mass formula?