  In this note, we consider arbitrary finite-dimensional real algebras
containing a copy of complex numbers. It is proved that matrices with entries
from an arbitrary finite-dimensional real algebra containing a square root of
negative one in its left (resp. right) associate set have left (resp. right)
eigenvalues. A quick consequence of our main result is the existence of left
and right eigenvalues for matrices with entries from finite-dimensional
alternatives algebras containing a copy of complex numbers, e.g., octonions,
and more generally matrices with entries from the real Cayley-Dickson algebras.
