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

laliste := {{1,(sqrt(2)*e7 + e1 - e2 - e3 - e4 - e5 - e6)/2},
{2,e1 + e2},
{3, - e1 + e2},
{4, - e2 + e3},
{5, - e3 + e4},
{6, - e4 + e5},
{7, - e5 + e6},
{8,e1 + e3},
{9,e1 + e4},
{10,e1 + e5},
{11,e1 + e6},
{12,e2 + e3},
{13,e2 + e4},
{14,e2 + e5},
{15,e2 + e6},
{16,e3 + e4},
{17,e3 + e5},
{18,e3 + e6},
{19,e4 + e5},
{20,e4 + e6},
{21,e5 + e6},
{22, - e1 + e3},
{23, - e1 + e4},
{24, - e1 + e5},
{25, - e1 + e6},
{26, - e2 + e4},
{27, - e2 + e5},
{28, - e2 + e6},
{29, - e3 + e5},
{30, - e3 + e6},
{31, - e4 + e6},
{32,sqrt(2)*e7},
{33,(sqrt(2)*e7 - e1 + e2 + e3 + e4 + e5 + e6)/2},
{34,(sqrt(2)*e7 + e1 - e2 + e3 + e4 + e5 + e6)/2},
{35,(sqrt(2)*e7 + e1 + e2 - e3 + e4 + e5 + e6)/2},
{36,(sqrt(2)*e7 + e1 + e2 + e3 - e4 + e5 + e6)/2},
{37,(sqrt(2)*e7 + e1 + e2 + e3 + e4 - e5 + e6)/2},
{38,(sqrt(2)*e7 + e1 + e2 + e3 + e4 + e5 - e6)/2},
{39,(sqrt(2)*e7 - e1 - e2 - e3 + e4 + e5 + e6)/2},
{40,(sqrt(2)*e7 - e1 - e2 + e3 - e4 + e5 + e6)/2},
{41,(sqrt(2)*e7 - e1 - e2 + e3 + e4 - e5 + e6)/2},
{42,(sqrt(2)*e7 - e1 - e2 + e3 + e4 + e5 - e6)/2},
{43,(sqrt(2)*e7 - e1 + e2 - e3 - e4 + e5 + e6)/2},
{44,(sqrt(2)*e7 - e1 + e2 - e3 + e4 - e5 + e6)/2},
{45,(sqrt(2)*e7 - e1 + e2 - e3 + e4 + e5 - e6)/2},
{46,(sqrt(2)*e7 - e1 + e2 + e3 - e4 - e5 + e6)/2},
{47,(sqrt(2)*e7 - e1 + e2 + e3 - e4 + e5 - e6)/2},
{48,(sqrt(2)*e7 - e1 + e2 + e3 + e4 - e5 - e6)/2},
{49,(sqrt(2)*e7 + e1 - e2 - e3 - e4 + e5 + e6)/2},
{50,(sqrt(2)*e7 + e1 - e2 - e3 + e4 - e5 + e6)/2},
{51,(sqrt(2)*e7 + e1 - e2 - e3 + e4 + e5 - e6)/2},
{52,(sqrt(2)*e7 + e1 - e2 + e3 - e4 - e5 + e6)/2},
{53,(sqrt(2)*e7 + e1 - e2 + e3 - e4 + e5 - e6)/2},
{54,(sqrt(2)*e7 + e1 - e2 + e3 + e4 - e5 - e6)/2},
{55,(sqrt(2)*e7 + e1 + e2 - e3 - e4 - e5 + e6)/2},
{56,(sqrt(2)*e7 + e1 + e2 - e3 - e4 + e5 - e6)/2},
{57,(sqrt(2)*e7 + e1 + e2 - e3 + e4 - e5 - e6)/2},
{58,(sqrt(2)*e7 + e1 + e2 + e3 - e4 - e5 - e6)/2},
{59,(sqrt(2)*e7 - e1 + e2 - e3 - e4 - e5 - e6)/2},
{60,(sqrt(2)*e7 - e1 - e2 + e3 - e4 - e5 - e6)/2},
{61,(sqrt(2)*e7 - e1 - e2 - e3 + e4 - e5 - e6)/2},
{62,(sqrt(2)*e7 - e1 - e2 - e3 - e4 + e5 - e6)/2},
{63,(sqrt(2)*e7 - e1 - e2 - e3 - e4 - e5 + e6)/2}}$

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

The following list sum_of_positive_roots_list collects$

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

Time: 10 ms

Time: 25800 ms

sum_of_positive_roots_list:={{{1,3},59},
{{1,12},58},
{{1,13},57},
{{1,14},56},
{{1,15},55},
{{1,16},54},
{{1,17},53},
{{1,18},52},
{{1,19},51},
{{1,20},50},
{{1,21},49},
{{1,22},60},
{{1,23},61},
{{1,24},62},
{{1,25},63},
{{1,33},32},
{{2,4},8},
{{2,22},12},
{{2,23},13},
{{2,24},14},
{{2,25},15},
{{2,26},9},
{{2,27},10},
{{2,28},11},
{{2,39},35},
{{2,40},36},
{{2,41},37},
{{2,42},38},
{{2,60},58},
{{2,61},57},
{{2,62},56},
{{2,63},55},
{{3,4},22},
{{3,8},12},
{{3,9},13},
{{3,10},14},
{{3,11},15},
{{3,26},23},
{{3,27},24},
{{3,28},25},
{{3,34},33},
{{3,49},43},
{{3,50},44},
{{3,51},45},
{{3,52},46},
{{3,53},47},
{{3,54},48},
{{4,5},26},
{{4,13},16},
{{4,14},17},
{{4,15},18},
{{4,29},27},
{{4,30},28},
{{4,35},34},
{{4,43},40},
{{4,44},41},
{{4,45},42},
{{4,55},52},
{{4,56},53},
{{4,57},54},
{{4,59},60},
{{5,6},29},
{{5,8},9},
{{5,12},13},
{{5,17},19},
{{5,18},20},
{{5,22},23},
{{5,31},30},
{{5,36},35},
{{5,40},39},
{{5,46},44},
{{5,47},45},
{{5,52},50},
{{5,53},51},
{{5,58},57},
{{5,60},61},
{{6,7},31},
{{6,9},10},
{{6,13},14},
{{6,16},17},
{{6,20},21},
{{6,23},24},
{{6,26},27},
{{6,37},36},
{{6,41},40},
{{6,44},43},
{{6,48},47},
{{6,50},49},
{{6,54},53},
{{6,57},56},
{{6,61},62},
{{7,10},11},
{{7,14},15},
{{7,17},18},
{{7,19},20},
{{7,24},25},
{{7,27},28},
{{7,29},30},
{{7,38},37},
{{7,42},41},
{{7,45},44},
{{7,47},46},
{{7,51},50},
{{7,53},52},
{{7,56},55},
{{7,62},63},
{{8,23},16},
{{8,24},17},
{{8,25},18},
{{8,29},10},
{{8,30},11},
{{8,39},34},
{{8,43},36},
{{8,44},37},
{{8,45},38},
{{8,59},58},
{{8,61},54},
{{8,62},53},
{{8,63},52},
{{9,22},16},
{{9,24},19},
{{9,25},20},
{{9,31},11},
{{9,40},34},
{{9,43},35},
{{9,46},37},
{{9,47},38},
{{9,59},57},
{{9,60},54},
{{9,62},51},
{{9,63},50},
{{10,22},17},
{{10,23},19},
{{10,25},21},
{{10,41},34},
{{10,44},35},
{{10,46},36},
{{10,48},38},
{{10,59},56},
{{10,60},53},
{{10,61},51},
{{10,63},49},
{{11,22},18},
{{11,23},20},
{{11,24},21},
{{11,42},34},
{{11,45},35},
{{11,47},36},
{{11,48},37},
{{11,59},55},
{{11,60},52},
{{11,61},50},
{{11,62},49},
{{12,26},16},
{{12,27},17},
{{12,28},18},
{{12,29},14},
{{12,30},15},
{{12,39},33},
{{12,49},36},
{{12,50},37},
{{12,51},38},
{{12,61},48},
{{12,62},47},
{{12,63},46},
{{13,27},19},
{{13,28},20},
{{13,31},15},
{{13,40},33},
{{13,49},35},
{{13,52},37},
{{13,53},38},
{{13,60},48},
{{13,62},45},
{{13,63},44},
{{14,26},19},
{{14,28},21},
{{14,41},33},
{{14,50},35},
{{14,52},36},
{{14,54},38},
{{14,60},47},
{{14,61},45},
{{14,63},43},
{{15,26},20},
{{15,27},21},
{{15,42},33},
{{15,51},35},
{{15,53},36},
{{15,54},37},
{{15,60},46},
{{15,61},44},
{{15,62},43},
{{16,29},19},
{{16,30},20},
{{16,31},18},
{{16,43},33},
{{16,49},34},
{{16,55},37},
{{16,56},38},
{{16,59},48},
{{16,62},42},
{{16,63},41},
{{17,30},21},
{{17,44},33},
{{17,50},34},
{{17,55},36},
{{17,57},38},
{{17,59},47},
{{17,61},42},
{{17,63},40},
{{18,29},21},
{{18,45},33},
{{18,51},34},
{{18,56},36},
{{18,57},37},
{{18,59},46},
{{18,61},41},
{{18,62},40},
{{19,31},21},
{{19,46},33},
{{19,52},34},
{{19,55},35},
{{19,58},38},
{{19,59},45},
{{19,60},42},
{{19,63},39},
{{20,47},33},
{{20,53},34},
{{20,56},35},
{{20,58},37},
{{20,59},44},
{{20,60},41},
{{20,62},39},
{{21,48},33},
{{21,54},34},
{{21,57},35},
{{21,58},36},
{{21,59},43},
{{21,60},40},
{{21,61},39},
{{22,29},24},
{{22,30},25},
{{22,35},33},
{{22,49},40},
{{22,50},41},
{{22,51},42},
{{22,55},46},
{{22,56},47},
{{22,57},48},
{{23,31},25},
{{23,36},33},
{{23,49},39},
{{23,52},41},
{{23,53},42},
{{23,55},44},
{{23,56},45},
{{23,58},48},
{{24,37},33},
{{24,50},39},
{{24,52},40},
{{24,54},42},
{{24,55},43},
{{24,57},45},
{{24,58},47},
{{25,38},33},
{{25,51},39},
{{25,53},40},
{{25,54},41},
{{25,56},43},
{{25,57},44},
{{25,58},46},
{{26,31},28},
{{26,36},34},
{{26,43},39},
{{26,46},41},
{{26,47},42},
{{26,55},50},
{{26,56},51},
{{26,58},54},
{{26,59},61},
{{27,37},34},
{{27,44},39},
{{27,46},40},
{{27,48},42},
{{27,55},49},
{{27,57},51},
{{27,58},53},
{{27,59},62},
{{28,38},34},
{{28,45},39},
{{28,47},40},
{{28,48},41},
{{28,56},49},
{{28,57},50},
{{28,58},52},
{{28,59},63},
{{29,37},35},
{{29,41},39},
{{29,46},43},
{{29,48},45},
{{29,52},49},
{{29,54},51},
{{29,58},56},
{{29,60},62},
{{30,38},35},
{{30,42},39},
{{30,47},43},
{{30,48},44},
{{30,53},49},
{{30,54},50},
{{30,58},55},
{{30,60},63},
{{31,38},36},
{{31,42},40},
{{31,45},43},
{{31,48},46},
{{31,51},49},
{{31,54},52},
{{31,57},55},
{{31,61},63},
{{34,59},32},
{{35,60},32},
{{36,61},32},
{{37,62},32},
{{38,63},32},
{{39,58},32},
{{40,57},32},
{{41,56},32},
{{42,55},32},
{{43,54},32},
{{44,53},32},
{{45,52},32},
{{46,51},32},
{{47,50},32},
{{48,49},32}}$

Time: 10 ms

Time: 43590 ms  plus GC time: 70 ms

listesverif:={}$

Time: 0 ms

Property (P) OK$

Time: 0 ms