Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \in S(L^*)$ with degree $n$ exists one multilinear symmetric form in $n$-variables, $g:L\times\dots\times L\rightarrow\mathbb{C}$, such that $g(x,\dots,x)=f(x)$. In Enveloping Algebras, the author Dixmier states that the following are equivalent.
For every $ x_1,\dots,x_n \in L$ and $x \in L$ ad-nilpotent, we have $g([x,x_1],\dots,x_n)+\cdots+g(x_1,\dots,[x,x_n])=0$.
For every $\tau \in \mathbb{C}$, $ x_1,\dots,x_n \in L$ and $x \in L$ ad-nilpotent, $$g(exp(ad \tau x)x_1,\dots,exp(ad \tau x)x_n)=g(x_1,\dots, x_n)$$
To see that $2$ implies $1$ is sufficient seeing that the left-side in $2$ is a polynomial in $\tau$ whose derivative in $\tau = 0$ is the left-side in $1$.
How Can I prove that 1 implies $2$? Thanks in advance