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. 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