A Lagrangian with $E_8\times E_8$ symmetry for the standard model and pre-gravitation I. -- The bosonic Lagrangian, and a theoretical derivation of the weak mixing angle

Building on earlier work, we propose an elementary Lagrangian for the unification of the standard model with pre-gravitation, assumed to have an unbroken $E_8 \times E_8$ symmetry. The Lagrangian is patterned after the kinetic energy of a free particle in Newtonian dynamics, generalising it to the matrix-valued Lagrangian dynamics of a 2-brane on a split bioctonionic space. Symmetry breaking gives rise to the standard model quarks and leptons of three generations along with the known gauge interactions, and a novel right-handed counterpart $SU(3)_{grav}\times SU(2)_R \times U(1)_g$ identified as pre-gravitation. The goal of the present series of papers is to investigate if this pre-quantum, pre-spacetime matrix-valued Lagrangian dynamics can lead to the emergence of quantum field theory for the standard model, and classical general relativity, possibly with some additional corrections. In this paper we work out the bosonic part of the Lagrangian, and show how the anticipated 32 gauge bosons are recovered from the elementary Lagrangian. As an application, we show that the asymptotic low energy weak mixing angle $\theta_W$ is given by a solution of the equation $1 = (1/2)\sqrt{ \cos(\theta_{W}/2)}+ \sqrt {\sin(\theta_{W}/2)}$, yielding $\sin^2\theta_W \approx 0.24996$.