I just wanted to chime in to plug my own Ext calculator (a joint venture with Dexter Chua). You can see an interactive calculation of $\mathbb{R}\mathrm{P}^4\wedge\mathbb{R}\mathrm{P}^4$ here:

[Ext chart here.](https://hoodmane.github.io/rust_ext/?data=eJztWcuu2jAQ3fcrrrz2gtjuQ9l2V6lSdbc3KHKDQZbyoEm4oory7_WjrkqLYTIBFlxWOPb4zMQzx5MjBtKqQm-1qvuOpC_kWXVN-apasqSk69QPkpKvUteEEln0uqlJOpDPTd317a7ov3R-QpYb9b2VeS0rZTbIlarMhpXspXkaMjeR-_1dZryYKV1vd70ZZ2Sf7JOM0Iw0W7uWLO1w1_tla1o0ar0248QsbFQdNrGMjDS2zCzmuHQGUVcM5oo5V-ewKDhwcSpwAQmcQ52xc84AbyYo8JyEc3YEjU1Pb-TMGaZUovlj02shOf6ODFNX0VT_HRefkuhz73jdsjk8BGDgIKwrB354SgLqLFLv7B_2zENL9hxzOZ48BmYx_3fFMeTip-_haI1zzD0cOW8Ej3GFwnE0jxQdx1S4gOQOjMYhRQfuNlFCHNBrFhbH9cEouQSmnUTeEkHUCBbHtMxInQkMq6O3MoK20YpF0DZarwLHJUAmwX0MgDU3Mo75NABhzY1M4O6faGwILvFzV_rRMmSHpXPBsOdhCQw9onEhrrloXIjUBCxrvdalckLRLT1_E3lX5ebHoZodqtWFWVrLslN-xkrGIciSlNHwxZZyGnpVKmi4BtL3NHyEeQPXfryBY2MwsLYfaOgp3sCRwhu4ZAYDa_uRhjr3Bi5D3sAdSjCwtp9GM650WTftnevexS11b3JL3bt46N470b3gGr25sAVHdll9D1Kuc9EeyvVNKNcJ5ALkDowGUq7gfnFz5To3sodyvUflOoFLV1eul9SaAsfxhzp8c-rwjzB0MBbF6r3-59bPrnWte_W00pWRhUZVyfKpala70qyOZBzHdwPi78zfNnZYyX2-UptWmSe-MIC_AAQJMQ4)

You can interactively add Adams differentials and it will propagate them using a few basic rules, but you'll mostly need to work them out yourself.