Calculating the exponential of an upper triangular matrix

Find the matrix exponential $e^A$ for $$ A = \begin{bmatrix} 2 & 1 & 1\\ 0 & 2 & 1\\ 0 & 0 & 2\\ \end{bmatrix}$$.

I think we should use the proberty

If $AB = BA$ then $e^{A+B} = e^A e^B$.

We can use that

$$\begin{bmatrix} 2 & 1 & 1\\ 0 & 2 & 1\\ 0 & 0 & 2\\ \end{bmatrix} =\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix} +\begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{bmatrix}$$

Both matrices obviously commute. But I dont know how to calculate the exponential of

$$\begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{bmatrix}.$$

Could you help me?

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