I'm working on Lie Algebra at the moment and while everything I hear and read about it makes sense, I get stuck with some exercises (The question How to determine the matrix of adjoint representation of Lie algebra? helps me a bit, but it still doesn't give me the confidence about what I have). Here is one of the exersices, with my work for as far as I understand it (or think I do):
Let $\mathfrak{g}$ be defined by the following commutator relations: $[e_1,e_2]=e_3, [e_2,e_3]=e_1, [e_3,e_1]=e_2$ (We don't know anything else about $\mathfrak{g}$. Describe the adjoint representation, i.e. for each element $\xi=ae_1+be_2+ce_3$, find the matrix of the operator $\text{ad}_{\xi}$. First, we determine $[\xi,\mathfrak{g}]=[ae_1+be_2+ce_3,\mathfrak{g}]$.
\begin{align*}
[\xi,e_1] &= 0\cdot e_1 – be_3 + ce_2= ce_2-be_3\\
[\xi,e_2] &= ae_3 + 0\cdot e_2 – ce_1=-ce_1+ae_3\\
[\xi,e_3] &= -ae_2 + be_1 + 0\cdot e_3=be_1-ae_2\\
\end{align*}
The matrix operator would then become $M=\begin{pmatrix}
0 & -a & a\\
b & 0 & -b\\
-c & c & 0
\end{pmatrix}$
Prove that $\mathfrak{g}$ is isomorphic to $so(3)$. These are all skew-symmmetric matrices, i.e. those of the form $M=\begin{pmatrix}
0 & -a & b\\
a & 0 & -c\\
-b & c & 0
\end{pmatrix}$. So clearly something has gone wrong. Any hints and pointing out of errors is welcome.
Best Answer
Something went wrong when you were writinig down $M_\xi$. Note that the coordinates of the image of the $i$-th basis vector $e_i$ form the $i$-th column of $M$. As, for example $$ [\xi, e_1] = ce_2 - be_3 $$ the first column should read $(0, c, -b)^t$, altogether we have $$ M = \begin{pmatrix} 0 & -c & b\\ c & 0 & -a\\ -b & a & 0 \end{pmatrix} $$ which looks very skew-symmetric to me.