An algebraic basis gauge transformation is defined here as a transformation on the set of intrinsic Octonion Algebra basis elements. They are linear combinations of these structured such that a reversible bijection is produced between each indexed intrinsic basis element and the same index gauge basis element. The gauge transformation is required to map any orientation for original Octonion Algebra to a gauge basis with identical index matched orientation. Orientation is required to be a global and local gauge invariant.These gauge transformation matrices are found to be lower 7x7 block diagonal members of the group SO(7). Any global gauge transformed Octonion covariant derivative is form invariant with the intrinsic basis representation. Allowing local parametrization variation, fields in the physics sense are added to the still present form invariant content through addition of the covariant differentiation connection, whose general form derivation is provided.Subgroups of PSL(2,7) give two methods for creating Octonion algebraic basis gauge transformations. Both are shown to be expressible as circle group fibrations over the basic quad basis subspace defined for a choice of Quaternion subalgebra. The chosen subalgebra gauge basis components are then produced from the basic quad fibration by a process called basic quad algebraic completion.One method uses permutation subgroups of PSL(2,7) that leave one non-scalar basis element unchanged. This is shown to produce a gauge comparable to the direct product U(1)xU(1)xU(1). This method provides a smooth map between any of the four sets of Quaternion subalgebra basis triplets that exclude the unchanged basis element, and each of the other three. This gauges out a four-fold Octonion symmetry on basis element choices representing 3D axial (closed products) and polar (open products) vector types.The other method uses permutation subgroups of PSL(2,7) that leave the set of basis elements in one Quaternion subalgebra triplet intact. Here, half-angle 2-torus fibrations on the basic quad subspace embed a standard orthonormal whole-angle spherical-polar basis in the preserved subalgebra after algebraic completion. Half-angle 3-torus basic quad fibrations embed a whole angle Euler Angle basis in the preserved subalgebra after algebraic completion.A composition between any two algebraic basis gauge transformations is shown to produce a third, forming a group operation with closure on algebraic basis gauge types. A parallelism between this composition and fiber product structure is demonstrated.