The command DGGramSchmidt applies the usual Gram-Schmidt procedure to a list S = [X1, X2, ... Xm], of m independent vectors, forms or tensors (the forms or tensors must all be of the same type).  The procedure returns a list orthogonal vectors, forms or tensors  [Y1, Y2, ... Ym] such that

The inner products of differential forms and tensors are calculated using TensorInnerProduct.

The default assumption is that the metric g is positive-definite, that is, the signature is [+1, +1, ... +1].  In this case the length of each Yk is normalized to 1.  With method = "un-normalized" the Yk are left un-normalized.

If, for example, the signature is [+1, -1, -1, -1], then the length of Y1 normalized to 1 and the lengths of Y2, Y3, Y4  are normalized to -1.

If, for example, the signature is [+1, -1, -1, -1] and the vectors X1, X2  are both null, then the usual Gram-Schmidt procedure will fail. In this case one can set signature = [[1,-1], 1, 1] and  DGGramSchmidt will take Y1 to be a vector of non-zero length in the span of X1, X2.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form DGGramSchmidt(...) only after executing the command with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-DGGramSchmidt.