Preliminaries.
(See [1] for further details.)
Let $M$ be a compact connected $C^\infty$ Riemannian manifold.
We say that a list $\sigma_1,\ldots,\sigma_n$ ($n \in \mathbb{N}$) of $C^\infty$ vector fields on $M$ fulfils the Hörmander condition if, letting $L$ be the Lie-algebra generated by $\sigma_1,\ldots,\sigma_n$, we have that for all $x \in M$, $\{l(x):l \in L\}=T_xM$.
For any list $\sigma_1,\ldots,\sigma_n$ of $C^\infty$ vector fields on $M$ fulfilling the Hörmander condition, and any $C^\infty$ vector field $b$ on $M$, there exists a value $\lambda_{b;\sigma_1,\ldots,\sigma_n} \in \mathbb{R}$ [called the maximal Lyapunov exponent] such that the following holds: Letting $(\phi_t \colon M \to M)_{t \geq 0}$ be the stochastic solution flow on $M$ generated by the Stratonovich SDE
$$ dX_t \ = \ b(x_t) \, dt \ + \ \sum_{i=1}^n \sigma_i(X_t) \circ dW_t^i $$
with $(W_t^1,\ldots,W_t^n)_{t \geq 0}$ being an $n$-dimensional Wiener process, we have that for every $x \in M$,
$$ \frac{\log\|(\mathrm{d}\phi_t)_x\|}{t} \overset{\text{a.s.}}{\to} \lambda_{b;\sigma_1,\ldots,\sigma_n} \ \text{ as } t \to \infty. $$
The question.
Let $b$ be a $C^\infty$ vector field on $M$.
Does there necessarily exist a list $\sigma_1,\ldots,\sigma_n$ of $C^\infty$ vector fields on $M$, fulfilling the Hörmander condition, such that for every list $\tau_1,\ldots,\tau_m$ ($m \in \mathbb{N}$) of $C^\infty$ vector fields on $M$, $\lambda_{b;\sigma_1,\ldots,\sigma_n,\tau_1,\ldots,\tau_m}<0$?
[1] Peter H. Baxendale. “Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms”. Spatial stochastic processes. Vol. 19. Progr. Probab. Birkhäuser Boston, Boston, MA, 1991, pp. 189–218
