Please post your question to the LAMMPS mailing list, not to me.
You can indeed obtain (a1, a2, a3) by the formula if you know all six parameters, lx, ly, lz, xy, xz, and yz. If you don’t, you can derive them from a, b, c, alpha, beta, and gamma.
See Section 6.12 (http://lammps.sandia.gov/doc/Section_howto.html#howto_12) of the manual for more details.
Ray