Engel was born in Lugau, Saxony, as the son of a Lutheran pastor.

Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for much of his life.

Engel was the co-author, with Sophus Lie, of the three volume work Theorie der Transformationsgruppen (publ.

Engel was the editor of the collected works[1][2] of Sophus Lie with six volumes published between 1922 and 1937; the seventh and final volume was prepared for publication but appeared almost twenty years after Engel's death.

With Paul Stäckel he wrote a history of non-Euclidean geometry (Theorie der Parallellinien von Euklid bis auf Gauss, 1895).