In previous works, we have proven that there are local tetrads in four-dimensional curved Lorentzian spacetimes that can be written in terms of two kinds of local structures, the skeletons and the gauge vectors. These tetrads diagonalize locally and covariantly the stress–energy tensors for systems of differential equations of the Einstein–Maxwell kind in the Abelian electromagnetic case, or of the Einstein–Maxwell–Yang-Mills kind when non-Abelian Yang–Mills fields are included, along with suitable Yang–Mills stress–energy tensors. Under local Lorentz transformations, these new tetrads conserve their skeleton-gauge vector structure. In this short paper, we will prove that given any general unit orthogonal tetrad in spacetime, we will be able to construct a new tetrad in the skeleton-gauge vector form. We will prove a theorem stating that the original orthonormal tetrad can be constructed or reexpressed with this same local tetrad skeleton-gauge vector structures. The theorems proved in this paper will enable the demonstration of new results in group isomorphism theorems in the future.

在之前的工作中，我们已经证明了四维弯曲洛伦兹时空中存在局部四分体，可以用两种局部结构（骨架和规范向量）来表示。这些四分体对阿贝尔电磁情况下爱因斯坦-麦克斯韦类微分方程组的应力-能量张量进行局部和协变对角化，或者当包含非阿贝尔杨-米尔斯场时对爱因斯坦-麦克斯韦-杨-米尔斯类微分方程组进行局部和协变对角化，以及合适的杨-米尔斯应力-能量张量。在局部洛伦兹变换下，这些新的四分体保留了它们的骨架规格矢量结构。在这篇简短的论文中，我们将证明，给定时空中任何通用单位正交四分体，我们将能够以骨架规范向量形式构造一个新的四分体。我们将证明一个定理，表明原始正交四分体可以用相同的局部四分体骨架规范向量结构构建或重新表达。本文所证明的定理将使将来群同构定理的新结果得以论证。