Let $\pi$ be a complex representation of the compact connected Lie group $G$
(no need for semisimplicity here) on a (finite-dimensional) vector space $V$. We say that $\pi$ is of *real* type if it comes from a representation of $G$ on a real vector space by extension of scalars, and we say that $\pi$ is of *quaternionic* type if it comes from a representation of G on a quaternionic vector space by restriction of scalars.
If $\pi$ is neither of real type nor of quaternionic type, we say that $\pi$ is of *complex* type.

Let $\rho$ be a real irreducible representation of $G$ on a real vector space
$W$. By Schur’s lemma, the centralizer of $\rho(G)$ in $\mathrm{End}(W)$ is an associative real division algebra, thus, by Frobenius’ theorem, isomorphic to one of (a) $\mathbb R$, (b) $\mathbb H$ or (c) $\mathbb C$.

The relation between real and complex representations is, respectively, that:
(a) the complexification $\rho^c$ is irreducible (and we say that $\rho$ is absolutely irreducible) and $\rho^c = \pi$ is a representation of real type;
(b) the complexification $\rho^c$ is reducible and $\rho^c=\pi\oplus\pi$ where $\pi$ is an irreducible
representation of quaternionic type;
(c) the complexification $\rho^c$ is reducible and $\rho^c=\pi\oplus\pi^*$ where $\pi$ is an irreducible
representation of complex type and $\pi^*$ is not equivalent to $\pi$ (where $\pi^*$ denotes the induced representation on $V^*=\bar V$).

Also $\rho $ is a real form of $\pi$ in the first case ($\rho^c=\pi$), but
$\rho$ is $\pi$ viewed as a real representation in the other two cases
($\rho=\pi^r$).

Regarding the adjoint representation of a **compact** simple Lie group, it is always absolutely irreducible. Simplicity of the group is equivalent to irreducibility of the representation. Even in the semisimple case, admiting and invariant complex structure would mean that the Lie group is a complex
Lie group viewed as real, not possible due to the compactness.

Cartan's theory of real representations of semisimple Lie algebras is masterfully presented in modern form in

ESI Lectures in Mathematics and Physics

Arkady L. Onishchik (Yaroslavl State University, Russia)

Lectures on Real Semisimple Lie Algebras and Their Representations
ISBN print 978-3-03719-002-9, ISBN online 978-3-03719-502-4
DOI 10.4171/002
February 2004, 95 pages, softcover, 17 x 24 cm.
24.00 Euro

absolutely simpleanisotropic algebraic $\mathbb{R}$-group $\bf G$. It follows that the adjoint representation of $G$ isabsolutely irreducible, and therefore, it cannot be of complex type. $\endgroup$ – Mikhail Borovoi Feb 18 '17 at 20:05realgroup $G$. The adjoint representation of $G$ is clearly of complex type. Of course $G$ is not compact. $\endgroup$ – Mikhail Borovoi Feb 18 '17 at 21:17