The present note is an extension of a previous paper on the same subject. In this paper a concise proof was given of a theorem by Scheffers to the effect that if a linear associative algebra contains the quaternion algebra as a subalgebra, both having the same modulus, then it can be expressed as the direct product of that quaternion algebra and another algebra. It was also shown that this theorem could be generalised to the extent of substituting a matric quadrate algebra for the quaternion algebra. In the present paper the theorem is extended to certain other types of algebras.