In section 2.6 of the notes by Natario, he uses Cartan's magic formula $$\mathcal{L}_V\omega = i_V(d\omega) + d(i_V\omega)$$to compute the second fundamental form of timelike hypersurface $\sigma = \sigma_0$ in $$g_1 = -d\tau^2 + a^2(\tau) \big[ d\sigma^2 +\sigma^2(d\theta^2 + \sin^2\theta d\varphi^2) \big]$$
This surface has unit normal $\frac{1}{a}\frac{\partial}{\partial \sigma}$.
I am confused however about how to compute $\mathcal{L}_n(a^2(\tau)d\sigma^2)$ term because it seems to give $-\dot{a}\ d\tau d\sigma$, but this term does not appear in claimed second fundamental form $K = a(\tau)\sigma_0 (d\theta^2 + \sin^2\theta d\varphi^2)$.
I am sure I have done something foolish, and would be very grateful to be pointed in the right direction!
$\frac{1}{2}\mathcal{L}_n (a^2 d\sigma^2) = a(\mathcal{L}_na)d\sigma^2 + a^2 (\mathcal{L}_nd\sigma)d\sigma\\ = a^2 d(d\sigma(n))d\sigma\\ = a^2 d( \frac{1}{a(\tau)})d\sigma\\$