A remark on Higman's result about separable algebras (Q1082399)

According to a result due to \textit{D. G. Higman} [Can. J. Math. 7, 509-515 (1955; Zbl 0065.260)] a finite dimensional k-algebra A is separable iff there exist dual k-bases \(\{a_ i\}\) and \(\{b_ i\}\) such that \(\sum b_ ixa_ i\) is a unit for some \(x\in A\). The authors show that \(a_ i\) and \(b_ i\) can be rearranged here whereas the condition \(\sum a_ ixb_ i=1\) is not equivalent to the separability of A. The nondegeneracy of the trace form \(T_{A/k}\) is equivalent to the fact that \(\sum a_ ib_ i\) is a unit of A.