On the “Non-conservation of parity in weak interactions” Kostrikin and Manin, in their Linear Algebra and Geometry, state that:


(The excerpt is on pp. 42-43.)
The statement comes after a proof of general linear group over reals having two connected components, and the definition of the orientation of a real vector space.
Is there an account for the non-conservation of parity in weak interactions accessible to mathematicians who know very little physics (such as myself)? An abstract mathematical interpretation of the Wu experiment would be for instance an admissible (actually the ideal) answer. As per the context this has something to do with orientation, but I'd like to have a clearer idea.
With what little physics I know I could not decipher the expositions that show up first (including T.D. Lee's 1957 Nobel speech). Finally let me note that after looking at some literature I convinced myself that possibly Weyl's 1929 paper "Elektron und Gravitation. I." is relevant, though I am not quite sure.

Note: I posted this question originally on M.SE at https://math.stackexchange.com/q/3975751/169085. It did not get much attention and upon reflection I thought it might be more suited to MO.
 A: I don't think the mathematics needs to be more involved than to appreciate the difference between an axial vector (or pseudovector) and a polar vector. An axial vector does not change sign upon inversion, while a polar vector does change sign. Angular momentum is an axial vector, linear momentum is a polar vector. If some physical process depends on the angle between angular momentum and linear momentum, then inversion modifies that process, so it is not invariant under the parity operation.
In the Wu experiment the axial vector is the nuclear spin and the polar vector is the momentum of electrons emitted by that nucleus. The experiment detected a dependence of the emission probability on the angle between the nuclear spin and the electron momentum, thereby demonstrating that the physical process responsible for the electron emission (the weak interaction) is not invariant under the parity operation.
A: If some interaction acts differently on the right-handed and left-handed components of a particle, the interaction is automatically parity violating. So you just need to look at the Lagrangian and notice the asymmetry. Or equivalently, if you see any Dirac $\gamma^5$ matrix or Levi-Civita symbols $\epsilon^{\mu\nu\rho\sigma}$ it means that the interaction on right and left components is treated differently.
I don't know of any reference on the Standard Model of Particle Physics suited for mathematicians, because those are topics that rely heavily on experimental results, and also the notation and conventions on Lie groups and algebras differ a lot from the math literature. But still, you can look for example in the book on Non-commutative Geometry by Alain Connes; he has a little subsection at very end of the book where he explains more or less what is the SM of PP about, and you can see the Lagrangian with each term explained.
As stressed before, parity violation not only occurs in the weak sector of the SM, but you can write down many field theories with that peculiarity. For example, you can construct a gauge theory choosing some gauge group, and some quantum fields representing the particle content that transform in representations of the group. A Dirac fermion field (like the electron, or proton inside the atomic nuclei) is represented (in four dimensional space-time) by a four-component spinor $\Psi$, and at the same time can be split in terms of two (two component) left-handed Weyl fields $\chi$ and $\xi$ as
$$
\Psi=\binom{\chi}{\xi^\dagger}.
$$
If $\Psi$ is in a representation $\mathsf{R}$ of the gauge group, then $\chi$ and $\xi^\dagger$ must be as well. Equivalently, $\chi$ must be in the representation $\mathsf{R}$, and $\xi$ must be the complex conjugate representation $\overline{\mathsf{R}}$. Now suppose that we have a single left-handed Weyl field $\psi$ in a complex
representation  $\mathsf{R}$. Such a gauge theory is automatically parity violating
because the right-handed Hermitian conjugate, $\psi^\dagger$, is in $\overline{\mathsf{R}}$, an inequivalent representation of the gauge group, and is said to be chiral. The Lagrangian can be written as
$$
\mathcal{L}=i\psi^\dagger \overline{\sigma}^\mu\partial_\mu \psi+\cdots
$$
with the ellipses denoting interactions and kinetic terms for other field, and $\overline{\sigma}^\mu$ a four vector formed with the Pauli matrices and the identity. This can also be written as
$$
\mathcal{L}=i\overline{\Psi}\gamma^\mu\partial_\mu P_\mathsf{L} \Psi+\cdots,
$$
with
$$
P_\mathsf{L}\Psi=\frac{1}{2}(1-\gamma^5)\binom{\psi}{0},
$$
and $\gamma^\mu$ the Dirac matrices. So you can see explicitly that the left-handed component is treated in a different way, that is, that there's some left-right asymmetry. For this to be parity invariant it should contain another identical term but with $P_\mathsf{R}$.

(Most chiral gauge theories do not exist as quantum field theories because of some subtle internal problems that QFT possesses, but still, there are certain groups for which those inconsistencies are not present.)
