Claim: When Metaclass II factorises into a double light cone, the null cone intersection is a line.

Define Lorentz null cones

AAtrans=(({{((b1^2)/ww), 0, 0, 0}, {0, -((b1^2)/ww), 0, 0}, {0, 0, - b1, 0}, {0, 0, 0, -b1}}));
BBtrans=(({{(((b1^2) (-2+ww))/(ww^2)), ((2 (b1^2))/(ww^2)), 0, 0}, {((2 (b1^2))/(ww^2)), -(((b1^2) (2+ww))/(ww^2)), 0, 0}, {0, 0, - b1, 0}, {0, 0, 0, -b1}}));

Suppose v is a null vector for both Lorentz metrics

v={a0,a1,a2,a3}

p0=v.AAtrans.v;
p1=v.BBtrans.v;

If p0 = 0 and p1 = 0, then it follows that p0-p1 = 0.

Simplify[p0-p1]

It follows that a0=a1

Simplify[{p0,p1}/.{a1->a0}]

It follows that a2=a3=0

Simplify[{p0,p1}/.{a0->a1,a2->0,a3->0}]
v/.{a0->a1,a2->0,a3->0}

Check

vec={t,t,0,0};

Simplify[vec.AAtrans.vec]
Simplify[vec.BBtrans.vec]