Koecher studied mathematics and physics at the Georg-August-Universität in Göttingen.
He qualified in 1954 at the Westfälische Wilhelms University in Münster.
From 1962 to 1970, Koecher was department chair at the University of Munich.
His main research area was the theory of Jordan algebras, where he introduced the Kantor–Koecher–Tits construction and the Koecher–Vinberg theorem.
He discovered the Koecher boundedness principle in the theory of Siegel modular forms.