alpha1 = {1,0}; alpha2 = {-1/2,Sqrt[3]/2}; alpha3 = alpha1+alpha2;
lambda1 = {1/2,Sqrt[3]/6}; lambda2 = {0,Sqrt[3]/3};

roots = Graphics[{RGBColor[1,0,0],Line[{-alpha1,alpha1}],
	Line[{-alpha2,alpha2}],Line[{-alpha3,alpha3}]}];

planes[n_] :=
Graphics[{RGBColor[0,1,0],Line[{-n lambda1, n lambda1}],
	Line[{-n lambda2, n lambda2}],
	Line[{-n(lambda1-lambda2), n(lambda1-lambda2)}]
}]

grid[n_] := 
Graphics[{RGBColor[0,0,1],
	Table[Line[{-n lambda1 + i lambda2, n lambda1 + i lambda2}],
	{i,-n,n}],
	Table[Line[{i lambda1 - n lambda2, i lambda1 + n lambda2}],
	{i,-n,n}]
}]

Show[grid[4],planes[5],roots,AspectRatio->Automatic]

s1[x_]:= x-2(x.alpha1)alpha1; s2[x_]:= x-2(x.alpha2)alpha2;
s21[x_]:=s2[s1[x]]; s12[x_]:=s1[s2[x]];s121[x_]:=s1[s2[s1[x]]];

weylorbit[x_] := Union[{x,s2[x],s21[x],s121[x],s12[x],s1[x]}];

weylline[x_] := Graphics[
	Line[{x,s2[x],s21[x],s121[x],s12[x],s1[x],x}]]

wtlines = {weylline[2lambda1+lambda2],
	weylline[2lambda2],weylline[lambda1]};

Show[grid[4],planes[5], wtlines, AspectRatio->Automatic]