I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. The thing is isotropic in $\mathbb Z,$ for example $19 + 45 = 64. $ Even if that were not the case, the orthogonal group would contain those for the binary forms $19 x^2 - z^2$ and $5 y^2 - z^2.$ Note that there is an answer for the original Pythagorean problem, http://mathoverflow.net/questions/76352/orthogonal-group-of-quadratic-form and to some extent http://en.wikipedia.org/wiki/Tree_of_Pythagorean_triples Oh, note that i found all null vectors of the form, http://math.stackexchange.com/questions/483496/pythagorean-triples-with-additional-parameters/483510#483510 which took two recipes because of an annoying odd even thing.
Anyway, given the diagonal matrix $F$ with diagonal entries $19,5,-1,$ here are a number of integral matrices $P$ that solve $P^t F P = F,$ and so are called automorphs or members of the (automorphism, orthogonal, isometry, rotation) group. As we are in odd dimension, we do not much care about the determinant and allow $\pm 1.$
Note also that a recent Bulletin included two articles on the integral orthogonal group for an indefinite quaternary form, one by Kontorovich and one by Fuchs, anyway see http://www.ams.org/journals/bull/2013-50-02/
Um, let's see, oddity, you can take all entries of $P$ positive if you wish, so I'm just printing those. Since we can negate any row or any column at our leisure, it hurts nothing.
QUESTION: can someone please give generators for the entire group, that is every isometry can be written as a word in said generators? I'm hoping I've given enough information to do that. I'm guessing it should be no more than about five matrices, with any luck in this list or evident products of same.
EDIT: Reading Keith's eight-page note at http://www.math.uconn.edu/~kconrad/blurbs/ called Orthogonal Group of $x^2 + y^2 - z^2.$ Maybe I can do this myself, given time. Not sure. I did learn a bit about reflections for the "Euclidean Forms" project.
LIST:
-1 0 0
0 1 0
0 0 1
1 0 0
0 -1 0
0 0 1
1 0 0
0 1 0
0 0 -1
1 0 0
0 1 0
0 0 1
1 0 0
0 9 4
0 20 9
1 0 0
0 161 72
0 360 161
39 10 10
38 11 10
190 50 49
39 110 50
38 101 46
190 530 241
39 290 130
38 299 134
190 1430 641
39 1990 890
38 1829 818
190 9590 4289
39 40 20
152 151 76
380 380 191
39 40 20
152 161 80
380 400 199
39 10 10
418 101 106
950 230 241
39 110 50
418 1211 550
950 2750 1249
39 10 10
1102 299 286
2470 670 641
170 0 39
0 1 0
741 0 170
170 780 351
0 9 4
741 3400 1530
210 140 79
76 49 28
931 620 350
210 320 151
76 119 56
931 1420 670
210 20 49
532 49 124
1501 140 350
210 20 49
1216 119 284
2869 280 670
360 250 139
494 341 190
1919 1330 740
360 530 251
494 731 346
1919 2830 1340
360 130 101
950 341 266
2641 950 740
550 480 249
228 201 104
2451 2140 1110
550 660 321
228 271 132
2451 2940 1430
550 60 129
1824 201 428
4731 520 1110
609 380 220
76 49 28
2660 1660 961
609 980 460
76 119 56
2660 4280 2009
609 320 200
1216 641 400
3800 2000 1249
609 20 140
1444 49 332
4180 140 961
759 730 370
646 619 314
3610 3470 1759
759 830 410
646 709 350
3610 3950 1951
759 220 200
836 241 220
3800 1100 1001
780 250 211
38 11 10
3401 1090 920
780 1970 899
38 101 46
3401 8590 3920
780 10 179
950 11 218
4009 50 920
780 670 349
1102 949 494
4199 3610 1880
780 950 461
1102 1339 650
4199 5110 2480
1500 310 371
494 101 122
6631 1370 1640
1500 130 349
1178 101 274
7049 610 1640