To tell whether the representation is real or quaternionic, you need to look at its highest weight. The situation is as follows.

First of all, you need to ensure that the representation is self-dual. This happens if and only if the highest weight lies in some vector subspace of $\mathfrak{h}^*$. (For a lot of groups, this is just the whole space).

After this, you have two cases:

• For some groups, every self-dual representation is of real type.
• For the other groups, the highest weights of real representations form a sublattice of index $2$ in the lattice of all self-dual weights. Basically, to determine the type of a representation, you must add together some of the (integer) coordinates of its highest weight, and examine the parity of the sum. If the sum is even, the representation is real; it it is odd, the representation is quaternionic.

I have put together a concise table listing all the existing real forms of all simple Lie groups and giving a recipe for each. It can be found here: http://www.math.u-psud.fr/~smilga/Lie_group_tables/tables.html ; the relevant part is the next-to-last item. The last item can also be of interest: it contains some explicit pictures for low-rank groups.

To tell whether the representation is real or quaternionic, you need to look at its highest weight. The situation is as follows.

First of all, you need to ensure that the representation is self-dual. This happens if and only if the highest weight lies in some vector subspace of $\mathfrak{h}^*$. (For a lot of groups, this is just the whole space).

After this, you have two cases:

• For some groups, every self-dual representation is of real type.
• For the other groups, the highest weights of real representations form a sublattice of index $2$ in the lattice of all self-dual weights. Basically, to determine the type of a representation, you must add together some of the (integer) coordinates of its highest weight, and examine the parity of the sum. If the sum is even, the representation is real; it it is odd, the representation is quaternionic.

I have put together a concise table listing all the existing real forms of all simple Lie groups and giving a recipe for each. It can be found here: http://www.normalesup.org/~smilga/Lie_group_tables/tables.html ; the relevant part is the next-to-last item. The last item can also be of interest: it contains some explicit pictures for low-rank groups.

To complement Jeffrey's answer, I have put together a concise table allowing to determine whether any given representation is real or quaternionic by looking at its highest weight. It can be found here: http://www.math.u-psud.fr/~smilga/Lie_group_tables/tables.html ; the relevant part is the next-to-last item.

To tell whether the representation is real or quaternionic, you need to look at its highest weight. The situation is as follows.

First of all, you need to ensure that the representation is self-dual. This happens if and only if the highest weight lies in some vector subspace of $\mathfrak{h}^*$. (For a lot of groups, this is just the whole space).

After this, you have two cases:

• For some groups, every self-dual representation is of real type.
• For the other groups, the highest weights of real representations form a sublattice of index $2$ in the lattice of all self-dual weights. Basically, to determine the type of a representation, you must add together some of the (integer) coordinates of its highest weight, and examine the parity of the sum. If the sum is even, the representation is real; it it is odd, the representation is quaternionic.

I have put together a concise table listing all the existing real forms of all simple Lie groups and giving a recipe for each. It can be found here: http://www.math.u-psud.fr/~smilga/Lie_group_tables/tables.html ; the relevant part is the next-to-last item. The last item can also be of interest: it contains some explicit pictures for low-rank groups.