  By means of the rotational transformations of octonion coordinate systems,
the paper aims to explore the physical properties of conserved quantities
relevant to the vectorial magnitudes within the material media, revealing the
simultaneity of some conserved quantities in the electromagnetic and
gravitational fields. J. C. Maxwell first utilized the algebra of quaternions
to describe the electromagnetic theory. The subsequent scholars studied the
physical properties of electromagnetic and gravitational fields simultaneously,
including the octonion linear momentum, angular momentum, torque and force
within the material media. According to the algebra of octonions, the scalar
parts of octonion physical quantities remain unchanged, while the vectorial
parts may alter, in the rotational transformations of octonion coordinate
systems. From the octonion characteristics, it is able to deduce a few
conserved quantities related to the vectorial magnitudes in the octonion space
$\mathbb{O}$, including the magnitudes of linear momentum, angular momentum,
torque and force. Similarly, it is capable of inferring several conserved
quantities relevant to the vectorial magnitudes in the transformed octonion
space $\mathbb{O}_u$, including the magnitudes of electric current, magnetic
moment and electric moment. Through the analysis and comparison, it is
concluded that some conserved quantities, relevant to the vectorial magnitudes,
are unable to be established simultaneously, from the point of view of the
octonion spaces. This is helpful to deepen the further understanding of some
conserved quantities related to the vectorial magnitudes.
