Proposition 7 in http://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/ contains a criterion (including a formula and proof; the proof is on page 131) to decide the type of the complex irreducible representation $\pi_\lambda$ of the compact connected semisimple Lie group $G$, where $\lambda$ is the highest weight, in terms of a maximal subset $\mathcal O=\{\beta_1,\ldots,\beta_\ell\}\subset\Delta^+$ of strongly orthogonal roots. Namely, $\pi_\lambda$ is unitary (not self-contragredient) if and only if $\lambda$ does not belong to the real span of $\mathcal O$. Otherwise it is symplectic (resp. orthogonal), meaning that it leaves a quaternionic (resp. real) structure invariant, if and only if $$ k(\lambda) = \sum_{i=1}^\ell \frac{(\beta_i,\lambda)-(\beta_i,s_0\lambda)}{(\beta_i,\beta_i)} $$ is an odd (resp. even) integer. Here $(,)$ is the inner product defined from the Killing form, and $$s_0 := s_{\beta_1}\cdots s_{\beta_\ell}$$ is the Weyl group element that maps the Weyl chamber to its negative.

The idea of the given proof is essentially to reduce the problem to a subgroup of $G$ isomorphic to the product of $\ell$ copies of $SU(2)$.

The integer $k(\lambda)$ has an interesting meaning. Let $\mathcal U^k(\mathfrak g_\mathbb C)$ be the k-th level in the natural filtration of the universal enveloping algebra of the complexified Lie algebra of $G$. Choose a highest weight vector $v_\lambda$ for $\pi_\lambda$. Then $s_0v_\lambda \in\mathcal U^{k(\lambda)}(\mathfrak g_\mathbb C)v_\lambda$ but $s_0v_\lambda\not\in\mathcal U^{k'}(\mathfrak g_\mathbb C)\lambda$ for any $k'<k(\lambda)$. Here
$$ s_0v_\lambda = \pi(X_{-\beta_1})^{n_1}\cdots\pi(X_{-\beta_\ell})^{n_\ell}v_\lambda$$ where $$n_i=\frac{(\beta_i,\lambda)-(\beta_i,s_0\lambda)}{(\beta_i,\beta_i)} $$ and the $X_{-\beta_i}$ are root vectors.

Proposition 7 in http://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/ contains a criterion (including a formula and proof; the proof is on page 131) to decide the type of the complex irreducible representation $\pi_\lambda$ of the compact connected semisimple Lie group $G$, where $\lambda$ is the highest weight, in terms of a maximal subset $\mathcal O=\{\beta_1,\ldots,\beta_\ell\}\subset\Delta^+$ of strongly orthogonal roots. Namely, $\pi_\lambda$ is unitary (not self-contragredient) if and only if $\lambda$ does not belong to the real span of $\mathcal O$. Otherwise it is symplectic (resp. orthogonal), meaning that it leaves a quaternionic (resp. real) structure invariant, if and only if $$ k(\lambda) = \sum_{i=1}^\ell \frac{(\beta_i,\lambda)-(\beta_i,s_0\lambda)}{(\beta_i,\beta_i)} $$ is an odd (resp. even) integer. Here $(,)$ is the inner product defined from the Killing form, and $$s_0 := s_{\beta_1}\cdots s_{\beta_\ell}$$ is the Weyl group element that maps the Weyl chamber to its negative.

The idea of the given proof is essentially to reduce the problem to a subgroup of $G$ isomorphic to the product of $\ell$ copies of $SU(2)$.

The integer $k(\lambda)$ has an interesting meaning. Let $\mathcal U^k(\mathfrak g_\mathbb C)$ be the k-th level in the natural filtration of the universal enveloping algebra of the complexified Lie algebra of $G$. Choose a highest weight vector $v_\lambda$ for $\pi_\lambda$. Then $s_0v_\lambda \in\mathcal U^{k(\lambda)}(\mathfrak g_\mathbb C)v_\lambda$ but $s_0v_\lambda\not\in\mathcal U^{k'}(\mathfrak g_\mathbb C)\lambda$ for any $k'<k(\lambda)$. Here
$$ s_0v_\lambda = \pi(X_{-\beta_1})^{n_1}\cdots\pi(X_{-\beta_\ell})^{n_\ell}v_\lambda$$ where $$n_i=\frac{(\beta_i,\lambda)-(\beta_i,s_0\lambda)}{(\beta_i,\beta_i)} $$ and the $X_{-\beta_i}$ are root vectors.

The table above lists the numbers $n_i$ for the fundamental representations of simple groups.