So let's say I have a three degrees-of-freedom robot with twists ${\xi}_{1}$, ${\xi}_2$, and ${\xi}_3$. The spatial Jacobian is given by $$ J = \begin{bmatrix}\xi_1 & Ad_{g1}{\xi}_2 & Ad_{g12}{\xi}_3\end{bmatrix} $$
I know that $$ Ad_{g1} = \begin{bmatrix}R_1 & p \times R_1\\ 0 & R_1\end{bmatrix} $$
However I am not sure how to calculate $Ad_{g12}$. Do I multiply $Ad_{g1} *Ad_{g2}$ or do I get the Transformation matrix of $\xi_1$ and $\xi_2$ and then use the formula for the adjoint?
You want to use the product of exponentials to calculate the transformation of $\zeta_1$ and $\zeta_2$ for $\theta_1$ and $\theta_2$.
To be more clear, using your notation of $g_{12}$: \begin{equation} g_{12} = e^{\hat{\zeta_1} \theta_1} \cdot e^{\hat{\zeta_2} \theta_2} \end{equation}
Where $\hat{\zeta_i}$ is the skew-symmetric matrix representation of the twist. Then you want to use the adjoint transformation of that resulting transformation matrix.
It is the transformation that takes the third joint's frame from its reference pose to it's current pose, and thus the adjoint transformation of this transforms the third joints unit twist to the current configuration.
