How to show isomorphism of two Lie Algebras?

Question

I need to show that Lie algebras so(3) and H with base {i, j, k} are isomorphic to each other.

Note that [i, j] = k, [j, k] = i, [k, i] = j

Answer 1

Let$$I=\begin{bmatrix}0&1&0\\-1&0&0\\0&0&0\end{bmatrix},\ J=\begin{bmatrix}0&0&-1\\0&0&0\\1&0&0\end{bmatrix}\text{, and }K=\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}.$$Then $I,J,K\in\mathfrak{so}(3)$ and$$[I,J]=K,[J,K]=I\text{, and }[K,I]=J.$$So…