L4a: A word about orientation.

 When you have a vector space $W$ with a metric and you want to talk about $\textrm{SO}(W)$ instead of $\textrm{O}(W)$. You do need an orientation, i.e., an element, up to positive multiples, of $\wedge^{\dim W} W$. Of course, this orientation is invisible when you look at them as matrices, since you are choosing a basis, say $\{w_i\}$, and the basis is giving you an orientation $w_1\wedge  \ldots \wedge w_n$. If you choose a different basis, say $\{-w_1\}\cup \{ w_2,\ldots,w_n \}$, you may get the opposite orientation, as it happens in this example: $-w_1\wedge \ldots \wedge w_n$.
 
What about $V\oplus V^*$? Apart from a canonical pairing, there is a canonical orientation on $V\oplus V^*$! We have to give an element of
$$\wedge^{(2\dim V)} (V\oplus V^*) = \wedge^{\dim V} V \otimes \wedge^{\dim V^*} V^*.$$ The canonical pairing between $\wedge^{\dim V} V$ and $\wedge^{\dim V^*} V^*$ gives that element. If you want, it is $1 \in \mathbb{R} \equiv \wedge^{\dim V} V \otimes \wedge^{\dim V^*} V^*,$
where the isomorphism is given by the canonical pairing.

We will talk about $\textrm{SO}(V\oplus V^*)$. Notice that the Lie algebra $\mathfrak{so}(V\oplus V^*)$ is exactly $\mathfrak{o}(V\oplus V^*)$, as it only depends on a neighbourhood of the identity element, which is the same for $\textrm{SO}$ and $\textrm{O}$.