  The paper aims to study some invariants and conservation laws relevant to
electromagnetic and gravitational fields, by means of the rotational
transformations of octonion coordinate systems. The scholars utilize the
octonions to analyze the electromagnetic and gravitational fields
simultaneously, including the octonion field potential, field strength, field
source, linear momentum, angular momentum, torque and force. When the octonion
coordinate system transforms rotationally, the vector part of one octonion may
alter, while the scalar part of the octonion will remain unchanged. This
property allows one to deduce a few invariants, such as the scalar part of
octonion radius vector, speed of light, and norm of octonion radius vector.
These invariants are the basic postulates for the Galilean transformation and
Lorentz transformation. Further, from the rotation transform of octonion
coordinate systems, it is capable of deducing several invariants, including the
mass, energy and power relevant to gravitational fields, in one octonion space
$\mathbb{O}$. And the term relevant to the electric charge will transform with
the rotation of coordinate systems. In another octonion space $\mathbb{O}_u$,
it is capable of inferring a few invariants, including the electric charge
related to electromagnetic fields. And the terms relevant to the mass and
energy will vary with the rotation of octonion coordinate systems. So the
invariants are divided into two different groups. In particular, the mass
conservation law and energy conservation law can be effective simultaneously.
But the charge conservation law and mass conservation law are unable to be
valid simultaneously, in the strict sense. It is beneficial to further
understand the laws of conservation.
