diff --git a/engine-proto/ConstructionViewer.jl b/engine-proto/ConstructionViewer.jl
index 29af212..c4845fa 100644
--- a/engine-proto/ConstructionViewer.jl
+++ b/engine-proto/ConstructionViewer.jl
@@ -4,6 +4,8 @@ using Blink
 using Colors
 using Printf
 
+using Main.Engine
+
 export ConstructionViewer, display!, opentools!, closetools!
 
 # === Blink utilities ===
@@ -133,7 +135,7 @@ mprod(v, w) =
 function display!(viewer::ConstructionViewer, elements::Matrix)
   # load elements
   elements_full = []
-  for elt in eachcol(elements)
+  for elt in eachcol(Engine.unmix * elements)
     if mprod(elt, elt) < 0.5
       elt_full = [0; elt; fill(0, 26)]
     else
@@ -206,7 +208,7 @@ end
 # ~~~ sandbox setup ~~~
 
 # in the default view, e4 + e5 is the point at infinity
-elements = sqrt(0.5) * BigFloat[
+elements = Engine.nullmix * sqrt(0.5) * BigFloat[
   1  1 -1 -1  0;
   1 -1  1 -1  0;
   1 -1 -1  1  0;
diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index dccb514..93102f8 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -39,11 +39,16 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
 
 # === elements ===
 
+## [temp] in light cone coordinates
+point(pos) = [pos; 1; dot(pos, pos)]
+
+## [temp] in standard coordinates
 plane(normal, offset) = [normal; offset; offset]
 
+## [temp] in standard coordinates
 function sphere(center, radius)
   dist_sq = dot(center, center)
-  return [
+  [
     center / radius;
     0.5 * ((dist_sq - 1) / radius - radius);
     0.5 * ((dist_sq + 1) / radius - radius)
@@ -52,8 +57,13 @@ end
 
 # === Gram matrix realization ===
 
+# basis changes
+nullmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [1 -1; 1 1]//2]
+unmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [1 1; -1 1]]
+
 # the Lorentz form
-Q = diagm([1, 1, 1, 1, -1])
+## [old] Q = diagm([1, 1, 1, 1, -1])
+Q = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [0 2; 2 0]]
 
 # project a matrix onto the subspace of matrices whose entries vanish at the
 # given indices
diff --git a/engine-proto/gram-test/circles-in-triangle.jl b/engine-proto/gram-test/circles-in-triangle.jl
index dd01ebf..ec6e1e0 100644
--- a/engine-proto/gram-test/circles-in-triangle.jl
+++ b/engine-proto/gram-test/circles-in-triangle.jl
@@ -55,7 +55,7 @@ gram = sparse(J, K, values)
 ## guess = Engine.rand_on_shell(fill(BigFloat(-1), 8))
 
 # set initial guess
-guess = hcat(
+guess = Engine.nullmix * hcat(
   Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
   Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)),
   Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
diff --git a/engine-proto/gram-test/overlapping-pyramids.jl b/engine-proto/gram-test/overlapping-pyramids.jl
index ee4c1fc..706a334 100644
--- a/engine-proto/gram-test/overlapping-pyramids.jl
+++ b/engine-proto/gram-test/overlapping-pyramids.jl
@@ -36,7 +36,7 @@ gram[1, 6] = gram[6, 1]
 # in this initial guess, the mutual tangency condition is satisfied for spheres
 # 1 through 5
 Random.seed!(50793)
-guess = hcat(
+guess = Engine.nullmix * hcat(
   sqrt(1/BigFloat(2)) * BigFloat[
     1  1 -1 -1  0;
     1 -1  1 -1  0;
diff --git a/engine-proto/gram-test/sphere-in-tetrahedron.jl b/engine-proto/gram-test/sphere-in-tetrahedron.jl
index 1d02d45..6bfb2c0 100644
--- a/engine-proto/gram-test/sphere-in-tetrahedron.jl
+++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl
@@ -41,7 +41,7 @@ gram = sparse(J, K, values)
 
 # set initial guess
 Random.seed!(99230)
-guess = hcat(
+guess = Engine.nullmix * hcat(
   sqrt(1/BigFloat(3)) * BigFloat[
     1  1 -1 -1  0
     1 -1  1 -1  0