Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ q(x) \neq 0 $ for all non zero $ x \in U $. Let us consider two quadratic forms $ q\_{1}$, $q\_{2} $, defined by
$$ q\_{1}(a,b,c,d) = ab + 2 ac - 2 ad - 2 bc + 2 bd - cd $$ and $$ q\_{2}(a,b,c,d) = 2 a^{2} - 13 ab + 4 ac + 4 ad + 2 b^{2} + 4 bc + 4 bd + 2 c^{2} - 13 cd + 2 d^{2} .$$
Then, what is the maximum dimension of a common anisotropic subspace of $ q\_{1} $ and $ q\_{2} $?



Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Dimension 4 is clearly impossible since your quadratic forms are isotropic, but dimension 3 is possible.


The following SageMath code generate random 3-dimensional subspaces and check whether they are simultaneously isotropic for both forms. It finds quite a lot of such subspaces, as an example you can take the subspace generated by the vectors


$$\pmatrix{2\cr0\cr0\cr1}, \pmatrix{-1\cr0\cr1\cr1}, \pmatrix{1\cr1\cr12\cr0}$$



```
M1 = Matrix([[ 0, 1, 2,-2],
             [ 1, 0,-2, 2],
             [ 2,-2, 0,-1],
             [-2, 2,-1, 0]])
M2 = Matrix([[  4,-13,  4,  4],
             [-13,  4,  4,  4],
             [  4,  4,  4,-13],
             [  4,  4,-13,  4]])
Q1 = QuadraticForm(QQ, M1)
Q2 = QuadraticForm(QQ, M2)

for k in range(50):

    basis = random_matrix(ZZ, 3, 4)

    N1 = matrix(QQ, 3, 3, lambda i, j: Q1.bilinear_map(basis[i], basis[j]))
    N2 = matrix(QQ, 3, 3, lambda i, j: Q2.bilinear_map(basis[i], basis[j]))

    q1 = QuadraticForm(QQ, N1)
    q2 = QuadraticForm(QQ, N2)
    
    if q1.anisotropic_primes() and q2.anisotropic_primes():
        print(basis)
        print()

```

