We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzian spacetimes and explore Weyl spinor fields as minimal left ideals within the complex quaternion algebra. The variational derivation of the coupled Einstein–Weyl equations from an action is presented, and the resulting field equations for both first and second order variations are derived and simplified. Exact plane symmetric solutions of the Einstein–Weyl equations are discussed, and two families of exact solutions describing left-moving and right-moving neutrino plane waves are provided. The study highlights the significance of adjusting a quartic self-coupling of the Weyl spinor in the action to ensure the equivalence of the field equations.

我们研究了 4 维洛伦兹时空中的复杂四元数值外微分形式，并探索了作为复杂四元数代数中的最小左理想的外尔旋量场。提出了耦合爱因斯坦-外尔方程从作用的变分推导，并推导并简化了所得到的一阶和二阶变化的场方程。讨论了爱因斯坦-外尔方程的精确平面对称解，并提供了描述左移和右移中微子平面波的两个精确解族。该研究强调了调整Weyl旋量的四次自耦合对于确保场方程等效性的重要性。