In this post I'll do a few very explicit computations for motivic cell structures of smooth projective toric varieties coming from the Białynicki-Birula decomposition, namely $\mathbb{P}^1, \mathbb{P}^1 \times \mathbb{P}^1, \mathbb{P}^2$ and Hirzebruch surfaces. It is a bit lengthy but maybe helpful to anyone who wants to do some explicit calculations with BB-decompositions. I hope you're accustomed to toric varieties, but I won't do anything fancy. You can safely skip the motivic part of this post.

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