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, $ and $76 + 5 = 81.$ 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, https://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, https://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. Alright, Keith gives his five reflections at the bottom of page 2. So, let's see. need to collect a bunch of vectors where the quadratic form evaluates to one of $-2,-1,1,2.$ Do that tomorrow, easy enough. List of some vectors with quadratic form equal to -2,-1,1,2 for 19 x^2 + 5 y^2 - z^2 x y z ----------------------------------------- -2 21 34 119 -2 21 170 391 -2 31 18 141 -2 39 110 299 -2 39 206 491 -2 59 186 489 -2 59 282 681 -2 131 270 831 -2 159 278 931 -2 201 94 901 -2 229 18 999 -2 231 70 1019 ----------------------------------------- -1 0 0 -1 -1 0 0 1 -1 0 4 9 -1 0 72 161 -1 1 1 5 -1 1 4 10 -1 1 11 25 -1 1 29 65 -1 1 76 170 -1 1 199 445 -1 4 8 25 -1 4 28 65 -1 4 172 385 -1 10 10 49 -1 10 106 241 -1 10 286 641 -1 14 10 65 -1 14 170 385 -1 15 4 66 -1 15 27 89 -1 15 39 109 -1 15 85 201 -1 15 113 261 -1 15 228 514 -1 15 300 674 -1 20 8 89 -1 20 76 191 -1 20 80 199 -1 20 284 641 -1 21 18 100 -1 21 23 105 -1 21 77 195 -1 21 87 215 -1 21 213 485 -1 21 238 540 -1 24 4 105 -1 24 84 215 -1 24 104 255 -1 25 1 109 -1 25 53 161 -1 25 56 166 -1 25 160 374 -1 25 167 389 -1 30 42 161 -1 30 266 609 -1 35 23 161 -1 35 46 184 -1 35 115 299 -1 35 161 391 -1 39 0 170 -1 39 85 255 -1 39 255 595 -1 45 15 199 -1 45 77 261 -1 45 122 336 -1 45 246 584 -1 49 11 215 -1 49 91 295 -1 49 124 350 -1 49 284 670 -1 50 46 241 -1 55 11 241 -1 55 104 334 -1 55 137 389 -1 71 8 310 -1 71 143 445 -1 71 167 485 -1 74 94 385 -1 79 28 350 -1 79 77 385 -1 79 133 455 -1 79 217 595 -1 85 53 389 -1 85 115 451 -1 85 274 716 -1 99 94 480 -1 99 99 485 -1 101 29 445 -1 101 91 485 -1 101 106 500 -1 101 179 595 -1 101 266 740 -1 105 77 489 -1 105 129 541 -1 111 15 485 -1 111 220 690 -1 111 265 765 -1 115 91 541 -1 115 134 584 -1 120 276 809 -1 125 94 584 -1 125 151 641 -1 129 87 595 -1 129 167 675 -1 130 134 641 -1 139 85 635 -1 139 190 740 -1 146 262 865 -1 151 56 670 -1 151 251 865 -1 154 182 785 -1 155 10 676 -1 155 106 716 -1 155 199 809 -1 171 77 765 -1 171 267 955 -1 174 186 865 -1 174 274 975 -1 179 133 835 -1 179 167 865 -1 179 182 880 -1 179 218 920 -1 185 29 809 -1 185 80 826 -1 185 293 1039 -1 190 218 961 -1 200 220 1001 -1 204 228 1025 -1 211 10 920 -1 211 115 955 -1 214 190 1025 -1 215 91 959 -1 220 4 959 -1 220 28 961 -1 221 143 1015 -1 221 293 1165 -1 224 124 1015 -1 225 57 989 -1 225 85 999 -1 230 122 1039 -1 245 137 1111 -1 249 104 1110 -1 251 151 1145 -1 251 179 1165 -1 275 77 1211 -1 276 76 1215 -1 295 8 1286 -1 295 167 1339 ----------------------------------------- 1 0 1 -2 1 0 1 2 1 0 17 38 1 10 9 48 1 10 111 252 1 10 273 612 1 18 7 80 1 18 257 580 1 28 39 150 1 28 249 570 1 30 31 148 1 60 7 262 1 72 191 530 1 78 1 340 1 78 199 560 1 80 159 498 1 98 225 660 1 118 297 840 1 168 193 850 1 172 9 750 1 180 199 902 1 210 193 1012 1 210 263 1088 1 228 121 1030 1 270 191 1252 1 282 73 1240 1 298 105 1320 ----------------------------------------- 2 3 0 13 2 3 52 117 2 7 16 47 2 7 44 103 2 13 16 67 2 13 124 283 2 17 28 97 2 17 32 103 2 17 124 287 2 17 136 313 2 27 16 123 2 27 92 237 2 27 120 293 2 37 28 173 2 37 104 283 2 37 196 467 2 43 76 253 2 47 32 217 2 57 180 473 2 57 272 657 2 87 24 383 2 87 180 553 2 93 60 427 2 93 196 597 2 147 24 643 2 163 280 947 2 173 52 763 2 173 76 773 2 193 44 847 2 197 92 883 2 197 260 1037 2 207 16 903 2 217 52 953 2 223 44 977 2 247 244 1207 2 277 52 1213 ----------------------------------------- x y z LIST of some orthogonal matrices, P solves P^t F P = F: -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