Keilhau made the first ascent of Falketind in 1820 along with two other climbers. He has later been labeled the discoverer of the mountain range of Jotunheimen. The mountain of Keilhaus topp is named after him. The 1820 journey, which he made along with fellow student Christian Boeck, was thoroughly documented in the article Nogle efterretninger om et hidtil ubekendt stykke af det söndenfjeldske Norge.