{c}

We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:
$$
R_z(\theta)
=
\begin{bmatrix}
cos(\theta) & -sin(\theta) & 0 \\
sin(\theta) & cos(\theta) & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
then we derive and evaluate at 0:
$$
L_z
=
\evalat{\dv{R_z(\theta)}{\theta}}{0}
=
\evalat{\begin{bmatrix}
-sin(\theta) & -cos(\theta) & 0 \\
cos(\theta) & -sin(\theta) & 0 \\
0 & 0 & 1 \\
\end{bmatrix}}{0}
=
\begin{bmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
$$
$L_z$ therefore represents the infinitesimal rotation.

Note that the <exponential map> reverses this and gives a finite rotation around the Z axis back from the <infinitesimal generator> $L_z$:
$$
e^{\theta L_z} = R_z(\theta)
$$

Repeating the same process for the other directions gives:
$$
L_x = TODO
L_y = TODO
$$
We have now found 3 <linearly independent> elements of the Lie algebra, and since $SO(3)$ has dimension 3, we are done.

Lie algebra of $SO(3)$

Lie algebra of SO(3)

Lie algebra of so 3

ID: lie-algebra-of-so-3