By Euler rotation Equation, torque $$\tau (t)=I \omega '(t) + \omega(t) \times I \omega(t)$$
where $I$ is the inertia tensor in rotating frame and is constant; $\omega(t)$ is the Angular velocity in the rotating frame, change with time; $\tau (t)$ is the torque in the rotating frame;
but we know that $$\tau (t) = L'(t) = (I \omega (t))' = I \omega '(t);$$ as $I$ is constant and where $L$ is the Angular momentum in the rotating frame;
$W(t)\times IW(t)=0$ always???
The Euler rotation equation, $$ \vec{\tau} = \mathbf{I} \dot{\vec{\omega}} + \vec{\omega} \times \left(\mathbf{I} \vec{\omega} \right), $$ implicitly refers to the description of the motion in a rotating reference frame, namely the reference frame rotating with the rigid body. In such a reference frame, it is no longer true that $d\vec{L}/dt = \vec{\tau}$.
Your second equation is a formula which is obtained for a certain reference frame.
On the other hand, your first one is relating two different frames.
The Euler rotation equation
$$ \vec{\tau} = \mathbf{I} \dot{\vec{\omega}} + \vec{\omega} \times \left(\mathbf{I} \vec{\omega} \right), \tag 1$$
with $\frac{d}{dt}L=\vec{\tau}$ and $L=\mathbf{I}\,\vec{\omega}$
but if $\vec{\tau}=0$ then $L=\mathbf{I}\,\vec{\omega}=const.\quad $
thus equation (1):
$$ \vec{0} = \vec{0} + \vec{\omega} \times \left(\mathbf{I} \vec{\omega} \right), \quad \surd$$
if the components of the angular momentum $L_b$,are given in body fixed coordinate system index $b$ then the time derivative of the angular momentum components $L_o$ in space system (index o) is:
$$\frac{d}{dt} L_o=\frac{d}{dt} L_b+\vec{\omega}\times L_b$$
now $\frac{d}{dt}L_o=\vec{\tau}=\vec{0}$ doesn't obtain that $\frac{d}{dt}L_b=\vec{0}$ thus $L_b$ is not conserved
remark:
the external torque $\vec{\tau}=\vec{r}\times \vec{F}$ is equal to zero if the force $\vec{F}$ is parallel to the radius of the rotation point $\vec{r}$, this force is call central force.
