Let $G$ be an $n$-dimensional Lie group over $\mathbb R$ (I am happy to assume $G$ is compact) and $(V_1,\dots,V_n$) a basis for its Lie algebra, $\mathfrak g$. Viewing the $V_i$ as left-invariant vector fields on $G$, they act linearly $C^{\infty}(G)\to C^{\infty}(G)$ by $$V_if(x):=\lim_{t\to 0}\frac{f(X_t^x)-f(x)}{t},$$ where $$\dot X_t^x=V_i(X_t^x), \quad X_0^x=x\in G.$$ We can then build another linear operator $L:C^{\infty}(G)\to C^{\infty}(G),\; L=\sum_i V_i^2$, where $V_i^2= V_i\circ V_i$.
Letting $\mu$ be a left Haar measure on $G$ and $f,g\in C_c^{\infty}(G)$ smooth functions of compact support, do we have the identity $$\int_GLf(x)g(x)\mu(dx)=\int_Gf(x)Lg(x)\mu(dx)?$$ I.e., is $L$ symmetric on $L^2(G,\mu)$? I've tried fiddling around with the above definitions, but seem only to arrive at massive, unrevealing formulae!