Sql server - comparison of all possible combinations and saving the result in another table

I have a simple sql query that asks if an object has the specified characteristics.
For simple example, I want to check if a group of objects is a double, triple, quadra color. And so I have variable characteristics (depending on the external user), in this example the user has chosen the colors: red ('C1'), yellow ('C2'), green ('C3'), blue ('C4'). And he wants to know if there are objects of red-yellow, red-green, red-blue ... red-yellow-green ... red-yellow-blue-green (any combination).

My code, in turn (in separate queries), looks for objects with common characteristics:

"SELECT [Object] FROM TableData WHERE [character A] = '" + C1 + "' INTERSECT SELECT [Object] FROM TableData WHERE [character A] = '" + C2 + "'";

next

"SELECT [Object] FROM TableData WHERE [character A] = '" + C1 + "' INTERSECT SELECT [Object] FROM TableData WHERE [character A] = '" + C3 + "'";

...

and then I have to save all found objects to a new table creating two new specific values for them

• how many colors the object is looking for
• category identifying colors (for example 'C12' - identifying red-yellow in a given case)

my code in C#

string sqlStatement = "INSERT INTO AnalysisTable (Colors, Object, level) VALUES (@val1,@val2,@val3)";
                            try
                            {
                                con.Open();
                                SqlCommand cmd = new SqlCommand(sqlStatement, con);
                                cmd.Parameters.Clear();
                                cmd.Parameters.AddWithValue("@val1", "C12");
                                cmd.Parameters.AddWithValue("@val2", C12[i].Object);
                                cmd.Parameters.AddWithValue("@val3", "2");
                                cmd.ExecuteNonQuery();
                            }

I check each combination one by one. It generates a lot of code and I'm sure it can be simplified somehow...
But i don't know how

Thanks in advance for any help

Ps. if object is red-yellow-green should save it four times as

1. red-yellow
2. red-green
3. yellow-green
4. red-yellow-green

Edit
simplified structure
https://dbfiddle.uk/8oUNkVdZ

In my attempts to simplify the code the problem is 'Subquery returned more than 1 value'. My core code gets around this because it fetches the values for a separate query first and then adds them. I couldn't find a workaround.

For a better explanation, the program code compares datasets after point specifying character.

May be this task better solved by reсursion. I will offer a simpler (direct) solution.
Try this. Temporary table not used.

with colors2 as(
  select c1,c2 from (values('white','black'),('white','green'),('white','red')
                       ,('black','green'),('black','red')
                       ,('green','red')
                    )t(c1,c2)
)
,colors3 as(
  select c1,c2,c3 from (values('white','black','green'),('white','black','red')
                          ,('white','green','red')
                          ,('black','green','red')
                    )t(c1,c2,c3)
)
,colors4 as(
  select c1,c2,c3,c4 from (values('white','black','green','red')
                    )t(c1,c2,c3,c4)
)

,t2 as(
select [object]
  ,string_agg([character A],',') colors
from TableData
group by [Object]
)
select object,c1,c2,'' c3,'' c4,colors  
from t2 t
left join colors2 t2 on charindex(t2.c1,t.colors)>0 and charindex(t2.c2,t.colors)>0
where c1 is not null
union all
select object,c1,c2,c3,'' c4,colors  
from t2 t
left join colors3 t3 on charindex(t3.c1,t.colors)>0 and charindex(t3.c2,t.colors)>0
     and charindex(t3.c3,t.colors)>0
where c1 is not null
union all
select object,c1,c2,c3,c4,colors  
from t2 t
left join colors4 t4 on charindex(t4.c1,t.colors)>0 and charindex(t4.c2,t.colors)>0
     and charindex(t4.c3,t.colors)>0 and charindex(t4.c4,t.colors)>0
where c1 is not null
order by [Object]

Query result

Pairs with null may be excluded simply.

Example, based on your fiddle