cannot allow (oneself) to live (with someone) under the canopy of heaven; cannot allow (the sworn enemy) to live in this world; irreconcilable (enemy)
不倶戴天の敵と呼ばれる関係が続いていた。
They continued a relationship called sworn enemies.
二人は不倶戴天の仲となり和解は困難だった。
The two became irreconcilable enemies, making reconciliation difficult.
不倶戴天の恨みを晴らす機会を窺った。
He looked for a chance to settle his irreconcilable grudge.