Clear[A];

A[1,1]=Table[RotateLeft[{(d^2),d},k],{k,0,1}]

A[1,2]=Table[RotateLeft[{0,b d},k],{k,0,1}]

A[2,1]=Transpose[A[1,2]]

A[2,2]=Table[RotateLeft[{(b^2)d,t b d},k],{k,0,1}]

(M=ArrayFlatten[Table[A[i,j],{i,1,2},{j,1,2}]])//MatrixForm

det=Collect[Det[M],t]

RootReduce[det/.Thread[{d,b,t}->{(1/2)Root[-64-32 #1+80 ((#1)^2)+32 ((#1)^3)-24 ((#1)^4)-6 ((#1)^5)+((#1)^6)&,6],Root[-1+17 #1-37 ((#1)^2)-48 ((#1)^3)+115 ((#1)^4)-22 ((#1)^5)+((#1)^6)&,2],Root[1+13349 #1-3190558 ((#1)^2)+42227282 ((#1)^3)-3008624 ((#1)^4)-304 ((#1)^5)+((#1)^6)&,2]}]]

t/.Solve[det==0,t]

Clear[A,b,d,t];

A[1,1]=Table[RotateLeft[{b (d^2),b d,0,0,b d},k],{k,0,4}]

A[1,2]=Table[RotateLeft[{0,t b d,0, (b^2)d,(b^2) d},k],{k,0,4}]

A[2,1]=Transpose[A[1,2]]

A[2,2]=Table[RotateLeft[{(b^3)d,(t^2)b d, t (b^2)d,t (b^2) d,(t^2)b d},k],{k,0,4}]

Unprotect[M]

(M=ArrayFlatten[Table[A[i,j],{i,1,2},{j,1,2}]])//MatrixForm

det=Collect[Det[M],t,Collect[#,b,Together]&]

TeXForm[det]

d=Root[-64-32 #1+80 #1^2+32 #1^3-24 #1^4-6 #1^5+#1^6&,6]

Table[N@Root[-64-32 #1+80 #1^2+32 #1^3-24 #1^4-6 #1^5+#1^6&,k],{k,1,6}]

Table[N@Root[-1+17 #1-37 #1^2-48 #1^3+115 #1^4-22 #1^5+#1^6&,k],{k,1,6}]

Table[N@Root[1+13349 #1-3190558 #1^2+42227282 #1^3-3008624 #1^4-304 #1^5+#1^6&,k],{k,1,6}]

SimpleRoots[G2]

{#,MatrixPresentation[G2][(Subscript[K, 1])][Irrep[G2][{1,0}],FundamentalBasis,#]}&/@Weights[G2,Irrep[G2][{1,0}]]

{#,MatrixPresentation[G2][(Subscript[K, 2])][Irrep[G2][{1,0}],FundamentalBasis,#]}&/@Weights[G2,Irrep[G2][{1,0}]]

{#,Together[MatrixPresentation[G2][(SuperMinus[(Subscript[X, 1])])][Irrep[G2][{1,0}],FundamentalBasis,#]]}&/@Weights[G2,Irrep[G2][{1,0}]]

{#,Together[MatrixPresentation[G2][(SuperMinus[(Subscript[X, 2])])][Irrep[G2][{1,0}],FundamentalBasis,#]]}&/@Weights[G2,Irrep[G2][{1,0}]]

{#,Together[MatrixPresentation[G2][(SuperPlus[(Subscript[X, 1])])][Irrep[G2][{1,0}],FundamentalBasis,#]]}&/@Weights[G2,Irrep[G2][{1,0}]]

{#,Together[MatrixPresentation[G2][(SuperPlus[(Subscript[X, 2])])][Irrep[G2][{1,0}],FundamentalBasis,#]]}&/@Weights[G2,Irrep[G2][{1,0}]]

MatrixPresentation[G2][(Subscript[K, 1])⊗\[ScriptOne]][Irrep[G2][{1,0}]⊗Irrep[G2][{1,0}],FundamentalBasis,{1,0}]

MatrixPresentation[G2][(Subscript[K, 2])⊗\[ScriptOne]][Irrep[G2][{1,0}]⊗Irrep[G2][{1,0}],FundamentalBasis,{1,0}]

b=Root[-1+17 #1-37 #1^2-48 #1^3+115 #1^4-22 #1^5+#1^6&,2]
t=Root[1+13349 #1-3190558 #1^2+42227282 #1^3-3008624 #1^4-304 #1^5+#1^6&,2]

RootReduce[det/.Thread[{d,b,t}->{(1/2)Root[-64-32 #1+80 ((#1)^2)+32 ((#1)^3)-24 ((#1)^4)-6 ((#1)^5)+((#1)^6)&,6],Root[-1+17 #1-37 ((#1)^2)-48 ((#1)^3)+115 ((#1)^4)-22 ((#1)^5)+((#1)^6)&,2],Root[1+13349 #1-3190558 ((#1)^2)+42227282 ((#1)^3)-3008624 ((#1)^4)-304 ((#1)^5)+((#1)^6)&,2]}]]

N[%]

det=Collect[Det[M]/.d->(b^2),t]

Solve[%==0,t]

Simplify[t/.%]

FusionAtlasPaths={"~/projects/fusionatlas/code/package/"};
$Path=$Path~Join~FusionAtlasPaths;
<<FusionAtlas`

δ=ReducedDimensionsByDepth[DnBigraph[14]][[-1,1]]

Collect[det/.b->Sqrt[δ],t,N]

Solve[%==0,t]