The preceding <Dynkin diagram> identifies the root system $D_3$ with $A_3$. The classification of finite-dimensional complex <Simple Lie algebras> by connected Dynkin diagrams therefore gives
$$
\boxed{\mathfrak{so}_6(\mathbb C)\cong\mathfrak{sl}_4(\mathbb C)}.
$$
This is the <Isomorphism between so6 and sl4>. Both algebras have dimension $15$, consistently with the classification.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / b / v / Solution

{c}
{title2=$\mathfrak{so}_6(\mathbb C)\cong\mathfrak{sl}_4(\mathbb C)$}

The root-system isomorphism $D_3\cong A_3$ and the classification of complex simple Lie algebras give $\mathfrak{so}_6(\mathbb C)\cong\mathfrak{sl}_4(\mathbb C)$.


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