What is the group $SO^*(12)$? Table 5 of this preprint refers to a group $SO^\star(12)$ but gives no reference for the name. Does anybody have a reference on it?
 A: The notation $\textrm{SO}^*(2 \ell)$ sometimes denotes the (semisimple, real, connected) subgroup of matrices in $\textrm{SO}(2 \ell, \Bbb C)$ that preserve a skew-Hermitian form on the standard representation, $\Bbb C^{2\ell}$. One can always choose a (complex) basis $(z_1, \ldots, z_{2 \ell})$ with respect to which the form is
$$-z_1 \overline w_{\ell + 1} + z_{\ell + 1} \overline w_1 - \cdots - z_\ell \overline w_{2\ell} + z_{2\ell} \overline w_{\ell} ,$$
and as a matrix group, written with respect to such a basis, $\textrm{SO}^*(2\ell)$ comprises exactly the $2\ell \times 2 \ell$ block matrices
$$\pmatrix{A & B\\-\bar{B}&-\bar{A}}$$
for which $A$ is a skew complex matrix and $B$ is a Hermitian complex matrix.
These groups are respectively real forms of $D_{\ell} = \mathrm{SO}(2\ell, \Bbb C)$ and so have (real) dimension $\ell (2 \ell - 1)$; their behavior depends mildly on the parity of $\ell$. The groups with $\ell \leq 4$ are isogenous to more familiar groups, and in that sense the smallest "new" group in this family is $\mathrm{SO}^*(10)$.
$$\begin{array}{rl}
\hline
\mathfrak{so}^*(2) &\cong &\mathfrak{u}(1) \cong \mathfrak{so}(2) \\
\mathfrak{so}^*(4) &\cong &\mathfrak{su}(2) \oplus \mathfrak{su}(1, 1) \cong \mathfrak{so}(3) \oplus \mathfrak{so}(1, 2)\\
\mathfrak{so}^*(6) &\cong &\mathfrak{su}(1, 3)\\
\mathfrak{so}^*(8) &\cong &\mathfrak{so}(2, 6)\\
\hline
\end{array}$$
For more, see, e.g., Helgason's Differential Geometry and Symmetric Spaces, $\S$IX.4, which does not give this group a descriptive name. Harvey's Spinors and Calibrations, $\S$1, calls this group the skew-unitary group and denotes it by the (I think, unusual) notation $\textrm{SK}(\ell, \mathbb H)$. See also Quaternionic Orthogonal Group.
A: I found the answer here.
"The star indicates the projective group, that is the factor group of its center $\epsilon_n\cdot $ with n-th roots of unity $\epsilon_n$."
The orthogonal group is $SO(n)$, and the star group is $SO^*(n)$.
