g[a_,s_] := Graphics3D[{LineForm[Thickness[s]],
	Map[Line[{{0,0,0},#}]&,a]}]

h[a_,s_] := Graphics3D[{PointSize[s],
	Map[Point[#]&,a]}]

e[1] = {1,0,0}; e[2] = {0,1,0}; e[3] = {0,0,1};

a2plus = {e[1]-e[2], e[1]-e[3], e[2]-e[3]};
a2 = Union[a2plus, -a2plus];

Show[g[a2,.007],h[a2,.03], ViewPoint->{2.020, -1.251, 2.409}]

b3plus = {u[1]=e[1]-e[2],u[2]=e[1]-e[3],u[3]=e[2]-e[3],
          u[4]=e[1]+e[2],u[5]=e[1]+e[3],u[6]=e[2]+e[3],
          e[1], e[2], e[3]};
b3 = Union[b3plus, -b3plus];

Show[g[b3,.005],h[b3,0.03]]

c3plus = {e[1]-e[2],e[1]-e[3],e[2]-e[3],
          e[1]+e[2],e[1]+e[3],e[2]+e[3],
          2e[1], 2e[2], 2e[3]};
c3 = Union[c3plus, -c3plus];

Show[g[c3,.005],h[c3,.03]]

fc[1] = Polygon[ { 2e[1], 2e[2], 2e[3]}];
fc[2] = Polygon[ { 2e[1], 2e[3],-2e[2]}];
fc[3] = Polygon[ {-2e[1],-2e[2], 2e[3]}];
fc[4] = Polygon[ {-2e[1], 2e[2], 2e[3]}];
fc[5] = Polygon[-{ 2e[1], 2e[2], 2e[3]}];
fc[6] = Polygon[-{ 2e[1], 2e[3],-2e[2]}];
fc[7] = Polygon[-{-2e[1],-2e[2], 2e[3]}];
fc[8] = Polygon[-{-2e[1], 2e[2], 2e[3]}];

Show[Graphics3D[{fc[1],fc[2],fc[3],fc[4],fc[5],fc[6],
	fc[7],fc[8]}],h[c3,.03]]

alpha[1] = {1,0,0};
alpha[2] = {-1/2,Sqrt[3/4],0};
alpha[3] = {0,-Sqrt[1/3], Sqrt[2/3]};
a3plus = {v[1]=alpha[1],v[2]=alpha[2],v[3]=alpha[3],
v[4]=alpha[1]+alpha[2],v[5]=alpha[2]+alpha[3],
v[6]=alpha[1]+alpha[2]+alpha[3]};
a3 = Union[a3plus,-a3plus];

Show[g[a3,.005],h[a3,.03]]

f[1]  = Polygon[ { v[1], v[4], v[6]}];
f[2]  = Polygon[-{ v[1], v[4], v[6]}];
f[3]  = Polygon[ { v[3], v[5], v[6]}];
f[4]  = Polygon[-{ v[3], v[5], v[6]}];
f[5]  = Polygon[ {-v[1], v[2], v[5]}];
f[6]  = Polygon[-{-v[1], v[2], v[5]}];
f[7]  = Polygon[ { v[3],-v[4],-v[2]}];
f[8]  = Polygon[-{ v[3],-v[4],-v[2]}];

f[ 9] = Polygon[ { v[1], v[6], v[3],-v[2]}];
f[10] = Polygon[-{ v[1], v[6], v[3],-v[2]}];
f[11] = Polygon[ { v[4], v[6], v[5], v[2]}];
f[12] = Polygon[-{ v[4], v[6], v[5], v[2]}];
f[13] = Polygon[ { v[3], v[5],-v[1],-v[4]}];
f[14] = Polygon[-{ v[3], v[5],-v[1],-v[4]}];

Show[Graphics3D[{f[9],f[10],f[11],f[12],f[13],f[14]}],
	g[a3,.007],h[a3,.03]]

Show[Graphics3D[{f[1],f[2],f[3],f[4],f[5],f[6],f[7],f[8]}],
	g[a3,.007],h[a3,.03]]

Show[Graphics3D[{f[1],f[2],f[3],f[4],f[5],f[6],f[7],f[8],
	f[9],f[10],f[11],f[12],f[13],f[14]}],h[a3,.03]]

If ±e[1], ±e[2], and ±e[3] are removed from b3 (or c3) we get a3:

aa3plus = {u[1]=e[1]-e[2],u[2]=e[1]-e[3],u[3]=e[2]-e[3],
          u[4]=e[1]+e[2],u[5]=e[1]+e[3],u[6]=e[2]+e[3]};
aa3 = Union[aa3plus, -aa3plus];

fb[1] = Polygon[ { u[1], u[5], u[4], u[2]}];
fb[2] = Polygon[-{ u[1], u[5], u[4], u[2]}];
fb[3] = Polygon[ { u[4], u[6],-u[1], u[3]}];
fb[4] = Polygon[-{ u[4], u[6],-u[1], u[3]}];
fb[5] = Polygon[ { u[5], u[6],-u[2],-u[3]}];
fb[6] = Polygon[-{ u[5], u[6],-u[2],-u[3]}];

Show[Graphics3D[{fb[1],fb[2],fb[3],fb[4],fb[5],fb[6]}],
	g[aa3,.007],h[aa3,.03]]

Show[Graphics3D[{fb[1],fb[2],fb[3],fb[4],fb[5],fb[6]}],
	g[b3,.007],h[b3,.03]]