Here we explain how to use the function "zariskiclosure" for computing
the Zariski closure of a finitely generated rational matrix group.
The algorithm is explained in the paper

Computing the Zariski closure of a finitely generated matrix group,
arXiv:2505.02577 [math.GR]

This function is implemented in OSCAR. There are two files:

* multrel.jl - which contains a function written in the Julia part of
               OSCAR

* zarclos.g - which has the functions written in GAP.

The main function is

* zariskiclosure( mats )

Here mats is a list of invertible nxn matrices with entries in Q.
The function returns a list [ L, cc ] with two elements: L is a
matrix Lie algebra, and cc is a list of matrices. Here L is the Lie algebra
of the smallest algebraic group G in GL(n,C) containing the matrices mats.
Furthermore, cc is a list of matrices having one element of each component
of G.

Example:

julia> using Oscar
  ___   ___   ___    _    ____
 / _ \ / __\ / __\  / \  |  _ \  | Combining and extending ANTIC, GAP,
| |_| |\__ \| |__  / ^ \ |  ´ /  | Polymake and Singular
 \___/ \___/ \___//_/ \_\|_|\_\  | Type "?Oscar" for more information
o--------o-----o-----o--------o  | Documentation: https://docs.oscar-system.org
  S Y M B O L I C   T O O L S    | 1.7.0-DEV #master 3a79eeb 2025-12-17

julia> include("multrel.jl");

julia> GAP.prompt()
gap> Read("zarclos.g");
gap> mats:=  [ [ [ 0, 0, -1, 0, 0, 0, 0, 0 ], [ -1, 0, 0, -3, 0, 0, 0, 0 ], 
>       [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, -1, -3, 0, 0, 0, 0, 0 ], 
>       [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, -1, 0, 0, -3 ], 
>       [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, -1, -3, 0 ] ], 
>   [ [ 0, 0, 0, 0, -3, 9, -1, 3 ], [ 0, 0, 0, 0, -1, 3, 0, 0 ], 
>       [ 0, 0, 0, 0, 0, -3, 0, 1 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], 
>       [ -3, 9, -1, 3, 0, 0, 0, 0 ], [ -1, 3, 0, 0, 0, 0, 0, 0 ], 
>       [ 0, -3, 0, 1, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ] ] ];;
gap> z:= zariskiclosure( mats );
[ <Lie algebra of dimension 10 over Rationals>, 
  [ [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], 
          [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], 
          [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ], 
      [ [ 0, 0, 0, 0, -3, 9, -1, 3 ], [ 0, 0, 0, 0, -1, 3, 0, 0 ], [ 0, 0, 0, 0, 0, -3, 0, 1 ], 
          [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ -3, 9, -1, 3, 0, 0, 0, 0 ], [ -1, 3, 0, 0, 0, 0, 0, 0 ], 
          [ 0, -3, 0, 1, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ] ] ] ]
gap> LieSolvableRadical( z[1] );
<Lie algebra of dimension 0 over Rationals>
gap> SemiSimpleType(z[1]);
"B2"




Remark: the implementation is "experimental" and can certainly be dramatically
improved in a number of ways.