Chaining bias terms in backprob

let's say I have a few linear layers $$l_1 \dots l_n$$:

$$y=I(\dots I(IX + b_1) + b_2) \dots +b_n)$$

where $$n$$ is sufficiently large and $$I$$ is the (nonparametric) identity matrix. The gradient for $$b_n$$ is then simply $$\frac{\partial L}{y}$$, but I am very confused about the gradients for the other biases. For example for $$b_{n-1}$$. Isn't it: $$\frac{\partial L}{b_{n-1}}=\frac{\partial L}{l_{n-1}}\frac{l_{n-1}}{b_{n-1}}=(\frac{\partial L}{y} I^T)I=\frac{\partial L}{y}$$?

But this would mean that by chaining $$n$$ bias terms I would get massive bias updates each gradient descent step! ($$n$$ times the usual bias update...). It also seems to violate the distributive-law$$^1$$?

$$[1]$$: What I mean: $$y=I(\dots I(IX + b_1) + b_2) \dots +b_n) = X + \sum_{i=1}^n b_i = X + b$$ for $$\sum_{i=1}^n b_i =: b$$. Shouldn't the gradient be compatible with this reshuffling?