Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.

The classical Siegel's formula in geometry of numbers states that for $f\in L^1(\mathbb R^d)$, let $\Lambda$ be a unimodular lattice, and let

$$\hat{f}(\Lambda):= \sum_{v\in \Lambda\setminus 0} f(v).$$

Then $$\int_X \hat{f} d\mu = \int_{\mathbb R^d} f dv.$$

I wonder if there is such a generalization to this result:

For $g\in L^1((\mathbb R^d)^k)$, $1 \le k \le d$ and unimodular lattice $\Lambda$, let $\hat{g}^k(\Lambda):=\sum_{(v_1,\dots,v_k)\in \Lambda^k \setminus 0} g(v_1,\dots, v_k)$. I speculate there might be such a generalization of Siegel's mean value formula:

$$\int_X \hat{g}^k d\mu = \int_{\mathbb R^d} \dots \int_{\mathbb R^d} g(v_1,\cdots,v_k) dv_1 \cdots dv_k$$

Is this correct and proved somewhere in the literature? Or can anyone suggest a proof for it below if not too complicated from the classical Siegel's formula?