I am trying to construct an instance of 7 unit vectors with complex coefficients, $v_i=(a_i,b_i,c_i,d_i), \,\,\, i=0,\ldots,6$ that satisfy a number of constraints regarding their overlap (or Hermitian angle). With the definition of the inner product as $$ (v_i,v_j)\equiv a^*_i a_j+b^*_i b_j+c^*_i c_j+d^*_i d_j, $$ where $^*$ denotes the complex conjugate, the vectors are constrained to have the following inner products: $$ |(v_0,v_j)| = \sqrt{\frac{5}{6}} \,, \,\, j=1,\ldots,6, \notag \\ |(v_i,v_j)| = \frac{4}{5} \,, \,\, i,j\neq0, \notag \\ (v_i,v_i) = 1 $$ with the last one being the normalisation condition, i.e. the fact that they should be unit vectors. Note that the Hermitian angle, $\theta$, can be defined through $\cos(\theta)=|(v,w)|/\lVert v\rVert \lVert w\rVert=|(v,w)|$, for unit vectors.
I can slightly simplify the problem by using the overall freedom to choose the first vector, that is, $v_0 = (1,0,0,0)$. Then, from the overlap of the other states with state $v_0$, we find that $|a_j|=\sqrt{\frac{5}{6}}, j=1,\ldots,6$. With this assumption, it follows that the condition $(v_i,v_i) = 1$ is equivalent to $$ |b_j|^2+|c_j|^2+|d_j|^2=1-|a_j|^2=\frac{1}{6} $$ or equivalently, in terms of real and complex parts $$ (b_j^{(R)})^2+(b_j^{(I)})^2+(c_j^{(R)})^2+(c_j^{(I)})^2+(d_j^{(R)})^2+(d_j^{(I)})^2=\frac{1}{6}\,, \,\, j=1,\ldots,6. $$ Now the last equation defines points on a 5-sphere in $\mathbb{R}^6$. It is known that there can exist $n+1$ equi-angular points on a sphere in $\mathbb{R}^n$; in the case $n=6$, it means there are at most 7. Since I am looking for 6 equi-angular points on a 5-sphere, a solution should exist to my problem.
This problem seems to be related to the construction of hypertetrahedra, or maybe $n$-simplices. I am not too familiar with them so I can't say for sure.
My questions are: are there any known solutions to this problem? Is there a simple way to get an explicit set of vectors satisfying these conditions?
Since approximate numerical solutions are also fine, I tried to obtain such a set numerically with Mathematica, which unfortunately runs forever. Thus, I am now doubting whether solutions even exist or not. Any help appreciated.