  The paper aims to explore the physical quantities of several invariants,
including the basic postulates of some types of crucial coordinate
transformations, conservation laws and continuity equations, in the
electromagnetic and gravitational fields. J. C. Maxwell first utilized the
quaternions to describe the electromagnetic theory. Subsequent scholars make
use of the octonions to study the physical properties of electromagnetic and
gravitational fields simultaneously, including the octonion field strength,
field source, angular momentum, torque and force. When an octonion coordinate
system transforms rotationally, the scalar part of one octonion will remain
unchanged, although the vector part of the octonion may alter. In the octonion
space $\mathbb{O}$ , some invariants can be derived from this octonion
property. A part of these invariants can be selected as the basic postulates of
Galilean transformation or Lorentz transformation. Similarly, it is able to
derive several invariants from the octonion property, in the transformed
octonion space $\mathbb{O}_u$ . And that the invariants can be chosen as the
basic postulates of a few new types of coordinate transformations. Further, the
combination of invariants in the octonion spaces can be applied as the basic
postulates of some new coordinate transformations, relevant to the norm of
physical quantities. Through the analysis, it is easy to find that each
conserved quantity has its preconditions from the perspective of octonion
spaces. This is helpful to deepen the further understanding of the physical
properties of conservation laws and other invariants.
