  We report a theoretical derivation of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix parameters and the accompanying mixing angles. These results are arrived
at from the exceptional Jordan algebra applied to quark states, and from
expressing flavor eigenstates (i.e. left-chiral states) as superposition of
mass eigenstates (i.e. the right-chiral states) weighted by square-root of
mass. Flavor mixing for quarks is mediated by the square-root mass eigenstates,
and the mass ratios used have been derived in earlier work from a left-right
symmetric extension of the standard model. This permits a construction of the
CKM matrix from first principles. There exist only four normed division
algebras, they can be listed as follows - the real numbers $\mathbb{R}$, the
complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$ and the octonions
$\mathbb{O}$. The first three algebras are fairly well known; however,
octonions as algebra are less studied. Recent research has pointed towards the
importance of octonions in the study of high energy physics. Clifford algebras
and the standard model are being studied closely. The main advantage of this
approach is that the spinor representations of the fundamental fermions can be
constructed easily here as the left ideals of the algebra. Also the action of
various Spin Groups on these representations too can be studied easily. In this
work, we build on some recent advances in the field and try to determine the
CKM angles from an algebraic framework. We obtain the mixing angle values as
$\theta_{12}=11.093^o, \theta_{13}=0.172^o, \theta_{23}=4.054^o$. In
comparison, the corresponding experimentally measured values for these angles
are $13.04^o \pm 0.05^o, 0.201^o \pm 0.011^o, 2.38^o \pm 0.06^o $. The
agreement of theory with experiment is likely to improve when running of quark
masses is taken into account.
