The 120 positive roots of E8.$

The following list collects {i,x(i)} where x(i) is the i-th positive root.$

laliste := {{1,(e1 - e2 - e3 - e4 - e5 - e6 - e7 + e8)/2},
{2,e1 + e2},
{3, - e1 + e2},
{4, - e2 + e3},
{5, - e3 + e4},
{6, - e4 + e5},
{7, - e5 + e6},
{8, - e6 + e7},
{9,e1 + e3},
{10,e1 + e4},
{11,e1 + e5},
{12,e1 + e6},
{13,e1 + e7},
{14,e1 + e8},
{15,e2 + e3},
{16,e2 + e4},
{17,e2 + e5},
{18,e2 + e6},
{19,e2 + e7},
{20,e2 + e8},
{21,e3 + e4},
{22,e3 + e5},
{23,e3 + e6},
{24,e3 + e7},
{25,e3 + e8},
{26,e4 + e5},
{27,e4 + e6},
{28,e4 + e7},
{29,e4 + e8},
{30,e5 + e6},
{31,e5 + e7},
{32,e6 + e7},
{33,x(33)},
{34,e6 + e8},
{35,e7 + e8},
{36, - e1 + e3},
{37, - e1 + e4},
{38, - e1 + e5},
{39, - e1 + e6},
{40, - e1 + e7},
{41, - e1 + e8},
{42, - e2 + e4},
{43, - e2 + e5},
{44, - e2 + e6},
{45, - e2 + e7},
{46, - e2 + e8},
{47, - e3 + e5},
{48, - e3 + e6},
{49, - e3 + e7},
{50, - e3 + e8},
{51, - e4 + e6},
{52, - e4 + e7},
{53, - e4 + e8},
{54, - e5 + e7},
{55, - e5 + e8},
{56, - e6 + e8},
{57, - e7 + e8},
{58,(e1 + e2 + e3 + e4 + e5 + e6 + e7 + e8)/2},
{59,( - e1 - e2 + e3 + e4 + e5 + e6 + e7 + e8)/2},
{60,( - e1 + e2 - e3 + e4 + e5 + e6 + e7 + e8)/2},
{61,( - e1 + e2 + e3 - e4 + e5 + e6 + e7 + e8)/2},
{62,( - e1 + e2 + e3 + e4 - e5 + e6 + e7 + e8)/2},
{63,( - e1 + e2 + e3 + e4 + e5 - e6 + e7 + e8)/2},
{64,( - e1 + e2 + e3 + e4 + e5 + e6 - e7 + e8)/2},
{65,(e1 - e2 - e3 + e4 + e5 + e6 + e7 + e8)/2},
{66,(e1 - e2 + e3 - e4 + e5 + e6 + e7 + e8)/2},
{67,(e1 - e2 + e3 + e4 - e5 + e6 + e7 + e8)/2},
{68,(e1 - e2 + e3 + e4 + e5 - e6 + e7 + e8)/2},
{69,(e1 - e2 + e3 + e4 + e5 + e6 - e7 + e8)/2},
{70,(e1 + e2 - e3 - e4 + e5 + e6 + e7 + e8)/2},
{71,(e1 + e2 - e3 + e4 - e5 + e6 + e7 + e8)/2},
{72,(e1 + e2 - e3 + e4 + e5 - e6 + e7 + e8)/2},
{73,(e1 + e2 - e3 + e4 + e5 + e6 - e7 + e8)/2},
{74,(e1 + e2 + e3 - e4 - e5 + e6 + e7 + e8)/2},
{75,(e1 + e2 + e3 - e4 + e5 - e6 + e7 + e8)/2},
{76,(e1 + e2 + e3 - e4 + e5 + e6 - e7 + e8)/2},
{77,(e1 + e2 + e3 + e4 - e5 - e6 + e7 + e8)/2},
{78,(e1 + e2 + e3 + e4 - e5 + e6 - e7 + e8)/2},
{79,(e1 + e2 + e3 + e4 + e5 - e6 - e7 + e8)/2},
{80,( - e1 - e2 - e3 - e4 + e5 + e6 + e7 + e8)/2},
{81,( - e1 - e2 - e3 + e4 - e5 + e6 + e7 + e8)/2},
{82,( - e1 - e2 - e3 + e4 + e5 - e6 + e7 + e8)/2},
{83,( - e1 - e2 - e3 + e4 + e5 + e6 - e7 + e8)/2},
{84,( - e1 - e2 + e3 - e4 - e5 + e6 + e7 + e8)/2},
{85,( - e1 - e2 + e3 - e4 + e5 - e6 + e7 + e8)/2},
{86,( - e1 - e2 + e3 - e4 + e5 + e6 - e7 + e8)/2},
{87,( - e1 - e2 + e3 + e4 - e5 - e6 + e7 + e8)/2},
{88,( - e1 - e2 + e3 + e4 - e5 + e6 - e7 + e8)/2},
{89,( - e1 - e2 + e3 + e4 + e5 - e6 - e7 + e8)/2},
{90,( - e1 + e2 - e3 - e4 - e5 + e6 + e7 + e8)/2},
{91,( - e1 + e2 - e3 - e4 + e5 - e6 + e7 + e8)/2},
{92,( - e1 + e2 - e3 - e4 + e5 + e6 - e7 + e8)/2},
{93,( - e1 + e2 - e3 + e4 - e5 - e6 + e7 + e8)/2},
{94,( - e1 + e2 - e3 + e4 - e5 + e6 - e7 + e8)/2},
{95,( - e1 + e2 - e3 + e4 + e5 - e6 - e7 + e8)/2},
{96,( - e1 + e2 + e3 - e4 - e5 - e6 + e7 + e8)/2},
{97,( - e1 + e2 + e3 - e4 - e5 + e6 - e7 + e8)/2},
{98,( - e1 + e2 + e3 - e4 + e5 - e6 - e7 + e8)/2},
{99,( - e1 + e2 + e3 + e4 - e5 - e6 - e7 + e8)/2},
{100,(e1 - e2 - e3 - e4 + e5 + e6 + e7 + e8)/2},
{101,(e1 - e2 - e3 + e4 - e5 + e6 + e7 + e8)/2},
{102,(e1 - e2 - e3 + e4 + e5 - e6 + e7 + e8)/2},
{103,(e1 - e2 - e3 + e4 - e5 - e6 + e7 + e8)/2},
{104,(e1 - e2 - e3 + e4 - e5 + e6 - e7 + e8)/2},
{105,(e1 - e2 - e3 + e4 + e5 - e6 - e7 + e8)/2},
{106,(e1 - e2 + e3 - e4 - e5 - e6 + e7 + e8)/2},
{107,(e1 - e2 + e3 - e4 - e5 + e6 - e7 + e8)/2},
{108,(e1 - e2 + e3 - e4 + e5 - e6 - e7 + e8)/2},
{109,(e1 - e2 + e3 + e4 - e5 - e6 - e7 + e8)/2},
{110,(e1 + e2 - e3 - e4 - e5 - e6 + e7 + e8)/2},
{111,(e1 + e2 - e3 - e4 - e5 + e6 - e7 + e8)/2},
{112,(e1 + e2 - e3 - e4 + e5 - e6 - e7 + e8)/2},
{113,(e1 + e2 - e3 + e4 - e5 - e6 - e7 + e8)/2},
{114,(e1 + e2 + e3 - e4 - e5 - e6 - e7 + e8)/2},
{115,( - e1 - e2 - e3 - e4 - e5 - e6 + e7 + e8)/2},
{116,( - e1 - e2 - e3 - e4 - e5 + e6 - e7 + e8)/2},
{117,( - e1 - e2 - e3 - e4 + e5 - e6 - e7 + e8)/2},
{118,( - e1 - e2 - e3 + e4 - e5 - e6 - e7 + e8)/2},
{119,( - e1 - e2 + e3 - e4 - e5 - e6 - e7 + e8)/2},
{120,( - e1 + e2 - e3 - e4 - e5 - e6 - e7 + e8)/2}}$

lalisteprime := {(e1 - e2 - e3 - e4 - e5 - e6 - e7 + e8)/2,
e1 + e2,
 - e1 + e2,
 - e2 + e3,
 - e3 + e4,
 - e4 + e5,
 - e5 + e6,
 - e6 + e7,
e1 + e3,
e1 + e4,
e1 + e5,
e1 + e6,
e1 + e7,
e1 + e8,
e2 + e3,
e2 + e4,
e2 + e5,
e2 + e6,
e2 + e7,
e2 + e8,
e3 + e4,
e3 + e5,
e3 + e6,
e3 + e7,
e3 + e8,
e4 + e5,
e4 + e6,
e4 + e7,
e4 + e8,
e5 + e6,
e5 + e7,
e6 + e7,
x(33),
e6 + e8,
e7 + e8,
 - e1 + e3,
 - e1 + e4,
 - e1 + e5,
 - e1 + e6,
 - e1 + e7,
 - e1 + e8,
 - e2 + e4,
 - e2 + e5,
 - e2 + e6,
 - e2 + e7,
 - e2 + e8,
 - e3 + e5,
 - e3 + e6,
 - e3 + e7,
 - e3 + e8,
 - e4 + e6,
 - e4 + e7,
 - e4 + e8,
 - e5 + e7,
 - e5 + e8,
 - e6 + e8,
 - e7 + e8,
(e1 + e2 + e3 + e4 + e5 + e6 + e7 + e8)/2,
( - e1 - e2 + e3 + e4 + e5 + e6 + e7 + e8)/2,
( - e1 + e2 - e3 + e4 + e5 + e6 + e7 + e8)/2,
( - e1 + e2 + e3 - e4 + e5 + e6 + e7 + e8)/2,
( - e1 + e2 + e3 + e4 - e5 + e6 + e7 + e8)/2,
( - e1 + e2 + e3 + e4 + e5 - e6 + e7 + e8)/2,
( - e1 + e2 + e3 + e4 + e5 + e6 - e7 + e8)/2,
(e1 - e2 - e3 + e4 + e5 + e6 + e7 + e8)/2,
(e1 - e2 + e3 - e4 + e5 + e6 + e7 + e8)/2,
(e1 - e2 + e3 + e4 - e5 + e6 + e7 + e8)/2,
(e1 - e2 + e3 + e4 + e5 - e6 + e7 + e8)/2,
(e1 - e2 + e3 + e4 + e5 + e6 - e7 + e8)/2,
(e1 + e2 - e3 - e4 + e5 + e6 + e7 + e8)/2,
(e1 + e2 - e3 + e4 - e5 + e6 + e7 + e8)/2,
(e1 + e2 - e3 + e4 + e5 - e6 + e7 + e8)/2,
(e1 + e2 - e3 + e4 + e5 + e6 - e7 + e8)/2,
(e1 + e2 + e3 - e4 - e5 + e6 + e7 + e8)/2,
(e1 + e2 + e3 - e4 + e5 - e6 + e7 + e8)/2,
(e1 + e2 + e3 - e4 + e5 + e6 - e7 + e8)/2,
(e1 + e2 + e3 + e4 - e5 - e6 + e7 + e8)/2,
(e1 + e2 + e3 + e4 - e5 + e6 - e7 + e8)/2,
(e1 + e2 + e3 + e4 + e5 - e6 - e7 + e8)/2,
( - e1 - e2 - e3 - e4 + e5 + e6 + e7 + e8)/2,
( - e1 - e2 - e3 + e4 - e5 + e6 + e7 + e8)/2,
( - e1 - e2 - e3 + e4 + e5 - e6 + e7 + e8)/2,
( - e1 - e2 - e3 + e4 + e5 + e6 - e7 + e8)/2,
( - e1 - e2 + e3 - e4 - e5 + e6 + e7 + e8)/2,
( - e1 - e2 + e3 - e4 + e5 - e6 + e7 + e8)/2,
( - e1 - e2 + e3 - e4 + e5 + e6 - e7 + e8)/2,
( - e1 - e2 + e3 + e4 - e5 - e6 + e7 + e8)/2,
( - e1 - e2 + e3 + e4 - e5 + e6 - e7 + e8)/2,
( - e1 - e2 + e3 + e4 + e5 - e6 - e7 + e8)/2,
( - e1 + e2 - e3 - e4 - e5 + e6 + e7 + e8)/2,
( - e1 + e2 - e3 - e4 + e5 - e6 + e7 + e8)/2,
( - e1 + e2 - e3 - e4 + e5 + e6 - e7 + e8)/2,
( - e1 + e2 - e3 + e4 - e5 - e6 + e7 + e8)/2,
( - e1 + e2 - e3 + e4 - e5 + e6 - e7 + e8)/2,
( - e1 + e2 - e3 + e4 + e5 - e6 - e7 + e8)/2,
( - e1 + e2 + e3 - e4 - e5 - e6 + e7 + e8)/2,
( - e1 + e2 + e3 - e4 - e5 + e6 - e7 + e8)/2,
( - e1 + e2 + e3 - e4 + e5 - e6 - e7 + e8)/2,
( - e1 + e2 + e3 + e4 - e5 - e6 - e7 + e8)/2,
(e1 - e2 - e3 - e4 + e5 + e6 + e7 + e8)/2,
(e1 - e2 - e3 + e4 - e5 + e6 + e7 + e8)/2,
(e1 - e2 - e3 + e4 + e5 - e6 + e7 + e8)/2,
(e1 - e2 - e3 + e4 - e5 - e6 + e7 + e8)/2,
(e1 - e2 - e3 + e4 - e5 + e6 - e7 + e8)/2,
(e1 - e2 - e3 + e4 + e5 - e6 - e7 + e8)/2,
(e1 - e2 + e3 - e4 - e5 - e6 + e7 + e8)/2,
(e1 - e2 + e3 - e4 - e5 + e6 - e7 + e8)/2,
(e1 - e2 + e3 - e4 + e5 - e6 - e7 + e8)/2,
(e1 - e2 + e3 + e4 - e5 - e6 - e7 + e8)/2,
(e1 + e2 - e3 - e4 - e5 - e6 + e7 + e8)/2,
(e1 + e2 - e3 - e4 - e5 + e6 - e7 + e8)/2,
(e1 + e2 - e3 - e4 + e5 - e6 - e7 + e8)/2,
(e1 + e2 - e3 + e4 - e5 - e6 - e7 + e8)/2,
(e1 + e2 + e3 - e4 - e5 - e6 - e7 + e8)/2,
( - e1 - e2 - e3 - e4 - e5 - e6 + e7 + e8)/2,
( - e1 - e2 - e3 - e4 - e5 + e6 - e7 + e8)/2,
( - e1 - e2 - e3 - e4 + e5 - e6 - e7 + e8)/2,
( - e1 - e2 - e3 + e4 - e5 - e6 - e7 + e8)/2,
( - e1 - e2 + e3 - e4 - e5 - e6 - e7 + e8)/2,
( - e1 + e2 - e3 - e4 - e5 - e6 - e7 + e8)/2}$

The following list sum_of_positive_roots_list collects$

{{i,j},k} such that x(i)+x(j) =x(k).$

Time: 30 ms

Time: 669490 ms  plus GC time: 90 ms

sum_of_positive_roots_list:={{{1,3},120},
{{1,15},114},
{{1,16},113},
{{1,17},112},
{{1,18},111},
{{1,19},110},
{{1,21},109},
{{1,22},108},
{{1,23},107},
{{1,24},106},
{{1,26},105},
{{1,27},104},
{{1,28},103},
{{1,36},119},
{{1,37},118},
{{1,38},117},
{{1,39},116},
{{1,40},115},
{{1,58},14},
{{1,59},46},
{{1,60},50},
{{1,61},53},
{{1,62},55},
{{1,63},56},
{{1,64},57},
{{2,4},9},
{{2,36},15},
{{2,37},16},
{{2,38},17},
{{2,39},18},
{{2,40},19},
{{2,41},20},
{{2,42},10},
{{2,43},11},
{{2,44},12},
{{2,45},13},
{{2,46},14},
{{2,59},58},
{{2,80},70},
{{2,81},71},
{{2,82},72},
{{2,83},73},
{{2,84},74},
{{2,85},75},
{{2,86},76},
{{2,87},77},
{{2,88},78},
{{2,89},79},
{{2,115},110},
{{2,116},111},
{{2,117},112},
{{2,118},113},
{{2,119},114},
{{3,4},36},
{{3,9},15},
{{3,10},16},
{{3,11},17},
{{3,12},18},
{{3,13},19},
{{3,14},20},
{{3,42},37},
{{3,43},38},
{{3,44},39},
{{3,45},40},
{{3,46},41},
{{3,65},60},
{{3,66},61},
{{3,67},62},
{{3,68},63},
{{3,69},64},
{{3,103},93},
{{3,104},94},
{{3,105},95},
{{3,106},96},
{{3,107},97},
{{3,108},98},
{{3,109},99},
{{4,5},42},
{{4,16},21},
{{4,17},22},
{{4,18},23},
{{4,19},24},
{{4,20},25},
{{4,47},43},
{{4,48},44},
{{4,49},45},
{{4,50},46},
{{4,60},59},
{{4,70},66},
{{4,71},67},
{{4,72},68},
{{4,73},69},
{{4,90},84},
{{4,91},85},
{{4,92},86},
{{4,93},87},
{{4,94},88},
{{4,95},89},
{{4,110},106},
{{4,111},107},
{{4,112},108},
{{4,113},109},
{{4,120},119},
{{5,6},47},
{{5,9},10},
{{5,15},16},
{{5,22},26},
{{5,23},27},
{{5,24},28},
{{5,25},29},
{{5,36},37},
{{5,51},48},
{{5,52},49},
{{5,53},50},
{{5,61},60},
{{5,66},65},
{{5,74},71},
{{5,75},72},
{{5,76},73},
{{5,84},81},
{{5,85},82},
{{5,86},83},
{{5,96},93},
{{5,97},94},
{{5,98},95},
{{5,106},103},
{{5,107},104},
{{5,108},105},
{{5,114},113},
{{5,119},118},
{{6,7},51},
{{6,10},11},
{{6,16},17},
{{6,21},22},
{{6,27},30},
{{6,28},31},
{{6,37},38},
{{6,42},43},
{{6,54},52},
{{6,55},53},
{{6,62},61},
{{6,67},66},
{{6,71},70},
{{6,77},75},
{{6,78},76},
{{6,81},80},
{{6,87},85},
{{6,88},86},
{{6,93},91},
{{6,94},92},
{{6,99},98},
{{6,101},100},
{{6,109},108},
{{6,113},112},
{{6,118},117},
{{7,8},54},
{{7,11},12},
{{7,17},18},
{{7,22},23},
{{7,26},27},
{{7,31},32},
{{7,38},39},
{{7,43},44},
{{7,47},48},
{{7,56},55},
{{7,63},62},
{{7,68},67},
{{7,72},71},
{{7,75},74},
{{7,79},78},
{{7,82},81},
{{7,85},84},
{{7,89},88},
{{7,91},90},
{{7,95},94},
{{7,98},97},
{{7,102},101},
{{7,105},104},
{{7,108},107},
{{7,112},111},
{{7,117},116},
{{8,12},13},
{{8,18},19},
{{8,23},24},
{{8,27},28},
{{8,30},31},
{{8,34},35},
{{8,39},40},
{{8,44},45},
{{8,48},49},
{{8,51},52},
{{8,57},56},
{{8,64},63},
{{8,69},68},
{{8,73},72},
{{8,76},75},
{{8,78},77},
{{8,83},82},
{{8,86},85},
{{8,88},87},
{{8,92},91},
{{8,94},93},
{{8,97},96},
{{8,104},103},
{{8,107},106},
{{8,111},110},
{{8,116},115},
{{9,37},21},
{{9,38},22},
{{9,39},23},
{{9,40},24},
{{9,41},25},
{{9,47},11},
{{9,48},12},
{{9,49},13},
{{9,50},14},
{{9,60},58},
{{9,80},66},
{{9,81},67},
{{9,82},68},
{{9,83},69},
{{9,90},74},
{{9,91},75},
{{9,92},76},
{{9,93},77},
{{9,94},78},
{{9,95},79},
{{9,115},106},
{{9,116},107},
{{9,117},108},
{{9,118},109},
{{9,120},114},
{{10,36},21},
{{10,38},26},
{{10,39},27},
{{10,40},28},
{{10,41},29},
{{10,51},12},
{{10,52},13},
{{10,53},14},
{{10,61},58},
{{10,80},65},
{{10,84},67},
{{10,85},68},
{{10,86},69},
{{10,90},71},
{{10,91},72},
{{10,92},73},
{{10,96},77},
{{10,97},78},
{{10,98},79},
{{10,115},103},
{{10,116},104},
{{10,117},105},
{{10,119},109},
{{10,120},113},
{{11,36},22},
{{11,37},26},
{{11,39},30},
{{11,40},31},
{{11,54},13},
{{11,55},14},
{{11,62},58},
{{11,81},65},
{{11,84},66},
{{11,87},68},
{{11,88},69},
{{11,90},70},
{{11,93},72},
{{11,94},73},
{{11,96},75},
{{11,97},76},
{{11,99},79},
{{11,118},105},
{{11,119},108},
{{11,120},112},
{{12,36},23},
{{12,37},27},
{{12,38},30},
{{12,40},32},
{{12,41},34},
{{12,56},14},
{{12,63},58},
{{12,82},65},
{{12,85},66},
{{12,87},67},
{{12,89},69},
{{12,91},70},
{{12,93},71},
{{12,95},73},
{{12,96},74},
{{12,98},76},
{{12,99},78},
{{12,118},104},
{{12,119},107},
{{12,120},111},
{{13,36},24},
{{13,37},28},
{{13,38},31},
{{13,39},32},
{{13,41},35},
{{13,57},14},
{{13,64},58},
{{13,83},65},
{{13,86},66},
{{13,88},67},
{{13,89},68},
{{13,92},70},
{{13,94},71},
{{13,95},72},
{{13,97},74},
{{13,98},75},
{{13,99},77},
{{13,118},103},
{{13,119},106},
{{13,120},110},
{{14,36},25},
{{14,37},29},
{{14,39},34},
{{14,40},35},
{{15,42},21},
{{15,43},22},
{{15,44},23},
{{15,45},24},
{{15,46},25},
{{15,47},17},
{{15,48},18},
{{15,49},19},
{{15,50},20},
{{15,65},58},
{{15,80},61},
{{15,81},62},
{{15,82},63},
{{15,83},64},
{{15,103},77},
{{15,104},78},
{{15,105},79},
{{15,115},96},
{{15,116},97},
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Time: 20 ms

Time: 525730 ms  plus GC time: 1690 ms

listesverif:={}$

Time: 0 ms

Property (P) OK$

Time: 0 ms