Poincaré and Galilei group - notation On this slide

it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of.
What does $\mathbb{R}^{1,3}$ mean?
Why does $\mathcal{G}$ have the extra $(\mathbb{R}^1 \times \mathbb{R}^3)$?
 A: $\mathbf{R}^{1,3}$ is Minkowski spacetime. It's a 4-dimensional real vector space with inner product \begin{align}x\cdot y = - x_0 y_0 + x_1 y_1 + x_2 y_2 + x_3 y_3.\end{align}
Note that in each case the decomposition is given in the form 
\begin{align}
\text{Group = (Boosts and Rotations) } \times \text{(Spacetime translations)}.
\end{align}
(Strictly speaking, this should be a semidirect product rather than a direct product, but feel free to ignore this detail.)
For the case of the Poincaré group $\mathcal{P}$, the Lorentz subgroup $SO(1,3)$ includes both rotations and boosts — boosts by themselves aren't a subgroup. But for the Galilei group, boosts (parameterized by a velocity vector in $\mathbf{R}^3$) and rotations ($SO(3)$) are independent.
The "extra" $(\mathbf{R}^1 \times \mathbf{R}^3)$ appearing in $\mathcal{G}$ is the group of non-relativistic spacetime translations, the analogue of $\mathbf{R}^{1,3}$ for $\mathcal{P}$. Again, for $\mathcal{G}$, space and time translations decompose.
