Idea : Create a new vector space in which $T$ is diagonalizable i.e. has 1D invariant subspaces. Pull back all the subspaces to $V$ via an appropriate map, wherein all the invariant subspaces get mapped back to the desired disjoint subspaces.
We do the following : I claim that $V$ can be given a vector space structure over the complex numbers (or a field isomorphic to the complex numbers).
More precisely, consider $\mathbb Q(T)$, the field created by taking all rational functions in $T$ with real coefficients (so elements of this field are matrices of the form $p(T)(q(T))^{-1}$ where $p,q$ are polynomials with real coefficients). One easily sees that this field is isomorphic to $\mathbb Q(i)$, by sending $x+iy \to xI + yT$ and noting the relation satisfied by $T$.
To make $V$ a vector space over $\mathbb Q(T)$ is obvious : addition is as usual and scalar multiplication is defined by $(aI+bT)v = av + b(Tv)$.
Denote $V_{\mathbb Q(T)}$ for $V$ with changed base field.
Once $V$ is an $\mathbb Q(T)$ subspace, the operator $T : V_{\mathbb Q(T)} \to V_{\mathbb Q(T)}$ (now scalar multiplication by the field element $T$) is obviously a diagonal operator regardless of basis, a multiple of the identity matrix. Consequently, we get that $V_{\mathbb Q(T)}$ breaks into $1$ dimensional subspaces which are invariant under $T$ (obviously, taking any basis, the elements of this basis are all eigenvectors, so form one dimensional invariant subspaces). This is similar to how scalar multiplication (in the usual setting) does not change matrix regardless of change of basis.
We now perform a restriction of scalars as follows. Define a map $i : V_{\mathbb R} \to V_{\mathbb Q(T)}$, by the identity map (!)
I claim that any one dimensional subspace $\overline{\{b\}}$ of $V_{\mathbb Q(T)}$ has a two dimensional preimage. But this is obvious : $\overline{\{b\}}$ in $\mathbb Q(T)$ consists of all the elements $(xI+yT)b = xb + yTb \in \overline{\{b,Tb\}_{\mathbb Q}}$, and $b,Tb$ are linearly independent over $\mathbb Q$. Furthermore this is $T$ invariant. (All easy checks)
Thus, if $b_1,...,b_n$ is a basis of $V_{\mathbb Q(T)}$, then $i^{-1} (\overline{\{b_j\}})$ is a two dimensional $T$ invariant subspace of $V_{\mathbb Q}$. Finally, these subspaces are disjoint, because their images under $i$ lie in different subspaces of $V_{\mathbb Q(T)}$.
It follows that $V = \oplus i^{-1}(b_j)$.
