Let $T:=(\mathbb{R}/\mathbb{Z})^r$ denote the rank $r$ compact torus. We know that all of its irreducible complex representations are one-dimensional given by the characters $$\chi_{k_1,\dots,k_r}(x_1,\dots,x_r):=\exp\left(2\pi i \sum_{i=1}^r k_ix_i\right)$$ for $k_1,\dots,k_r\in \mathbb{Z}$.
Goal: I would like to prove that any irreducible real representations of $T$ is either the trivial representation (corresponding to $k_i=0,\forall i$) or a two-dimensional representation, for $k_1,\dots,k_r\in \mathbb{Z}$ not all zero, given by $$ (x_1,\dots,x_r) \mapsto \begin{pmatrix} \cos(2\pi \sum_{i=1}^r k_ix_i) & \sin(2\pi \sum_{i=1}^r k_ix_i) \\ -\sin(2\pi \sum_{i=1}^r k_ix_i) & \cos(2\pi \sum_{i=1}^r k_ix_i)\end{pmatrix} $$ and let's denote it by $W_{k_1,\dots,k_r}$.
My attempt is that, given any nontrivial irreducible real representation $(\pi,V)$ of $T$, we can take its complexification $V\otimes \mathbb{C} \cong \mathbb{C}^{\dim_{\mathbb{R}}(V)}$ which is then a direct sum of complex characters $\chi_{k_1,\dots,k_r}$ with $k_1,\dots,k_r$ not all zero. Since $V$ is real, these nontrivial characters come up in conjugate pairs. Since $$ \begin{pmatrix} \cos(2\pi \sum_{i=1}^r k_ix_i) & \sin(2\pi \sum_{i=1}^r k_ix_i) \\ -\sin(2\pi \sum_{i=1}^r k_ix_i) & \cos(2\pi \sum_{i=1}^r k_ix_i)\end{pmatrix} \begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix} \begin{pmatrix} \exp(2\pi i \sum_{i=1}^r k_ix_i) & 0 \\ 0 & \exp(-2\pi i \sum_{i=1}^r k_ix_i)\end{pmatrix} $$ we see that $\chi_{k_1,\dots,k_r}\oplus \chi_{-k_1,\dots,-k_r} \cong_{\mathbb{C}} W_{k_1,\dots,k_r}\otimes \mathbb{C}$. So I wonder, does this show that $V\cong_{\mathbb{R}} W_{k_1,\dots,k_r}$ as real representations?
Question: Could anybody salvage the above argument, or give a new argument, to prove the classification of irreducible real representations of the compact torus $T$? Thanks in advance!