$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V\_{\mathbb{R}}$ over the field of real numbers. 

Now I have to show that if $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals. This is my attempt so far:

Since the order of $G$ is odd, it has no non-trivial elements whose order is 2. So by Frobenius-Schur we know that $1=\sum\_{V}\text{dim}(V)FS(V)$ where $FS(V)$ is the Frobenius-Schur indicator, this sum can be of course be written as $$1=\sum\_{V\text{ is of real type}}\text{dim}(V)-\sum\_{V\text{ is of quaternionic type}}\text{dim}(V).$$
Since $G$ is non-trivial, it has by the above equality an irreducible representation which is quaternionic and by the above lemma, this one is not realisable over the reals.

My question is whether my reasoning is not flawed, especially when I am using the lemma. Any help would be appreciated. Thanks.