A homotopy theoretic and homological proof is given to a theorem of H. Scheerer: If two compact simply connected Lie groups are homotopy equivalent they are isomorphic. Both the original and the present proofs make use of the known list of the simple Lie groups. These are distinguishable by their mod 2 cohomology. However, distinct products of simple groups may produce mod p equivalent spaces. This paper proves that the above phenomena cannot occur simultaneously at the primes 2 and 3. Thus, two simply connected compact Lie groups which are mod 2 and mod 3 equivalent are isomorphic.