  Bose-Einstein condensation of electron-hole pairs, exciton condensation, has
been effortfully investigated since predicted 60 years ago. Irrefutable
evidence has still been lacking due to experimental difficulties in verifying
the condensation of the charge neutral and non-magnetic spin-singlet excitons.
Whilst, condensation of spin-triplet excitons is a promising frontier because
spin supercurrent and spin-Seebeck effects will be observable. A canonical
cobaltite LaCoO$_{3}$ under very high magnetic fields is a propitious
candidate, yet to be verified. Here, we unveil the exotic phase diagram of
LaCoO$_{3}$ up to 600 T generated using the electromagnetic flux compression
method and the state-of-the-art magnetostriction gauge. We found the continuous
magnetostriction curves and a bending structure, which suggest the emergence of
two distinct spin-triplet exciton condensates. By constructing a
phenomenological model, we showed that quantum fluctuations of excitons are
crucial for the field-induced successive transitions. The spin-triplet exciton
condensation in a cobaltite, which is three-dimensional and thermally
equilibrated, opens up a novel venue for spintronics technologies with
spin-supercurrent such as a spin Josephson junction.
