The two commuting triples each generate a copy of the complexified <SU(2) Lie algebra>. Hence the <chiral decomposition of the complex Lorentz algebra> is
$$
\boxed{\mathfrak{so}(1,3)_\mathbb C
\cong\mathfrak{sl}(2,\mathbb C)\oplus
\mathfrak{sl}(2,\mathbb C)}.
$$
In physicists' compact notation this is
$\mathfrak{su}(2)_+\oplus\mathfrak{su}(2)_-$; the real Lorentz algebra relates the two factors by complex conjugation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 4 / vii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-302/4/vii/solution