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I suggest reading the result 6.2.17 in the book written by A.J.Hahn and O.T.O'Meara called "The Classical Groups and K-Theory". The result is a general version of the Cartan-Dieudonné-Scherk theorem (which implies the Cartan-Diuedonné theorem). The result is stated in the language of "Quadratic spaces over form rings". The case of the defective orthogonal groups is a particular case of the general "unitary groups" $U_n(V)$.