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The classical LLL algorithm [A. K. Lenstra et al., Math. Ann. 261, 515–534 (1982; Zbl 0488.12001)] inputs a symmetric nonsingular, definite Gram matrix $$G$$ of dimension $$n$$ and from that calculates a Gram matrix corresponding to the reduced basis. For non-singular matrices there are several extensions of the LLL algorithm, for example the modified LLL algorithm of M. Pohst [J. Symb. Comput. 4, 123–127 (1987; Zbl 0629.10001)]. D. Simon [Math. Comput. 74, No. 251, 1531–1543 (2005; Zbl 1078.11072); “Quadratic equation in dimension 4,5 and more”, Preprint (2005)] considers the LLL algorithm with indefinite matrices.