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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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Generators: x y z;
Parameter: t;
Limiting weight: 22;
Relations:

[[x,y],x] - 2 x;
[[x,y],y] + 2 y;

[y,z];
[[x,y],z] + 10 z;
[x,[x,[x,[x,[x,[x,[x,[x,[x,[x,[x,z]]]]]]]]]]];

[z,[x,z]];
[[x,z],[x,[x,z]]];
[[x,[x,z]],[x,[x,[x,z]]]] + 6 t z;
[[x,[x,[x,z]]],[x,[x,[x,[x,z]]]]] + 8 t [x,[x,z]];
[[x,[x,[x,[x,z]]]],[x,[x,[x,[x,[x,z]]]]]] + 8 t [x,[x,[x,[x,z]]]] - 6 t^2 y;