No. With the way Gmail, Yahoo, and other email providers have set up their email security, only emails sent out on their infrastructure will be authenticated properly. You would end up getting many bounces because you would look like an illegitimate sender.
You can use as many different sending domains or from: addresses as you would like. For example, if you have multiple brands represented by one Account Engagement account, you may use a different from: address or domain for each brand's newsletter. It is important that if you do this, you first set up email authentication (DomainKeys/DKIM) for each of the domains you intend to use. Note that links will still be tracked and rewritten using the primary tracker domain you designated, regardless of what your sending domain is.
Do not attempt to send emails from a domain you do not own (for example, on behalf of a client). There are a few reasons you shouldn't do this:
Do not use the exact same domain for your email sending domain and your tracker domain. For example, if your tracker domain is go.company.com, your email sending domain cannot also be go.company.com. Using identical domains for both features will likely result in authentication errors and deliverability troubles.
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