Special Case of Lie-algebra

Question

Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have to show that the first one is a special case of the second one (for $n=3$). I write out both $[\cdot\ ,\ \cdot]$, but then? I don't know how to show that this is a special case. Can someone help me?! Thanks

Answer 1

The following steps will lead to a solution:

1. $\mathfrak{so}(3)$ is the set of all $ 3 \times 3$ matrix $X$ such that $X + X^T = 0$. A basis for this space is given by matrices $E,F,G$ where $$E = \left(\begin{array}{ccc} 0 & 1 & 0 \\ -1& 0& 0 \\ 0 & 0 & 0 \end{array}\right), F =\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0& 0& 0 \\ -1 & 0 & 0 \end{array}\right),G = \left(\begin{array}{ccc} 0 & 0 & 0 \\0 & 0& 1 \\ 0 & -1 &0 \end{array}\right).$$
2. Compute commutator relations for the $E,F$ and $G$. What are $[E,F]$, $[F,G]$ and $[E,G]$?
3. Compute Commutator relations for the usual basis vectors $e_1,e_2,e_3$ under the cross product.
4. Decide whether or not to map $E$ to $e_1$ say, $F$ to $e_2$ say and $G$ to $e_3$.
5. Why is the map that you have now a Lie algebra isomorphism?