Wording of the exchange of rings between Catholic and non-Christian?

At the marriage between a Catholic and a Non Christian, is it necessary that the Non Christian party during the nuptial rite take the name of the Trinity, for example at the exchange of the rings. “take this ring…in the name of the Father and of the Son and of the Holy Spirit.” why cant the non christian party just use the word “God” instead of the above?

The Rite of Marriage states:

  1. If circumstances so require, the blessing and exchange of rings can be omitted.

If the non-Christian party has an objection to the wording of the exchange of rings then it should be omitted.

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